Under this head, we may give the first place to the improvement which has taken place in the culture of Sea-Cale; and of this we shall treat pretty fully.

Sea-Cale.

The cultivation usually recommended consisted merely in covering the shoots, at the approach of spring, to the depth of a few inches, with dry earth, or with sand or gravel, in order to the blanching and internerating of the shoots. These were cut as they appeared in March and April. Now, however, the blanching is not only much more completely effected, but simple means have been devised of supplying the table with shoots for half the year, including all the winter months. It has of late become a market vegetable, and appears plentifully on the stalls of Covent-Garden, and more sparingly on those of the Edinburgh green-market. It is somewhat remarkable, that, in regard to this excellent culinary article, we have decidedly anticipated our neighbours the French. The Manuel du Jardinier for 1807 speaks only of the leaves being used, and, justly enough, condemns them as coarse. In the recent editions of the Bon Jardinier (1818, 1819), the blanched shoots are at last recommended, and the English mode of culture is mentioned. But this mode of culture is not yet practised in the marais of Paris, and sea-cale shoots will still be looked for in vain in the celebrated marché aux herbes of that capital.

The practice of the best cultivators shall here be described.—It is considered proper that the sea-cale bed should be trenched at least two feet deep. The soil should be rather light, and should have a dry bottom. If manure be added, it ought to consist of sea-weed, or of tree-leaves well rotted; the shoots being very apt to imbibe a disagreeable flavour from recent dungs and coarse manures. The plant may be propagated by offsets, or by small pieces of the root, having eyes or buds attached to them; but it rises freely from the seed: this is sown in March, generally in patches of three or four seeds, placed four or five inches separate, leaving fully two feet between each patch. During the first two years, the chief things to be attended to are hoeing and weeding, and rejecting any superfluous plants, in case all the seeds may have germinated. At the approach of winter, some gardeners throw a little light stable dung over the whole surface of the bed: a covering of fresh sandy soil, to the depth of two inches, answers equally well. In the third year, the plants become fit for blanching; and if the sea-cale bed be judiciously managed, it will continue productive for several successive years. In order, however, to ensure a succession of young and vigorous plants, and to provide for the bad effects of forcing, which is generally destructive to the plants subjected to it, it is proper to sow a small bed of sea-cale yearly. Fresh seed may always be kept in readiness, by allowing two or three plants to produce their flowers and seeds each year; the flowers, which are white and smell of honey, appear in May, and are followed by the seeds in September.

Various modes of blanching the shoots have been resorted to. In the first volume of the Memoirs of the Caledonian Horticultural Society, Sir George Mackenzie describes a very convenient method. The sea-cale bed is merely covered, early in the spring, with clean and dry oat-straw, which is removed as often as it becomes wet or musty. The shoots rise through the straw, and are at the same time pretty well blanched. Mr Barton, formerly gardener at Bothwell Castle, employed tree-leaves for this purpose. When these naturally fell in the end of autumn, he caused them to be swept together, and laid over the sea-cale bed to the depth perhaps of two feet. He found that a thin covering of stable dung, sufficient only to keep the leaves from being blown about, was useful in forwarding the production of the sea-cale shoots, a slight fermentation being thus induced. The shoots rise sweet and tender among the leaves, in the early part of spring; but it must evidently be difficult in this way to regulate the heat of fermentation, and safer to avoid it. Another method practised by many gardeners consists in placing over each plant a flower-pot of the largest size, inverted; and blanching-pots, constructed for this express purpose, are described by Mr Maher in the first volume of the Transactions of the Horticultural Society of London. These have since been much improved, by fitting them with moveable lids, the utility of which will presently appear. Such pots, we may remark, should not be made to taper much at top; but should be nearly of equal width throughout, in order to give room for taking off such shoots as are ready, without injuring the others. It may be proper to provide from thirty to sixty pots; and it may be expected that each pot will, on an average, furnish a dish and a half of shoots during the season.

With the aid of these pots, sea-cale is now forced in a very simple way, in the open border. In the latter end of the autumn, a bed of vigorous sea-cale is dressed off, that is, the stalks are cut over, and all decayed leaves are removed. The ground is at the same time stirred or loosened around the plants, and a thin stratum of fine gravel or of sifted coal-ashes is laid on the surface, in order to keep down earth-worms. A pot with a moveable cover is placed over each plant, or over each patch of plants, if two or more have remained together. Stable litter is then closely packed all around the pots, and pressed firmly down; and successive quantities are added, till the pots be buried to the depth of a foot or more; the whole thus assuming the form and appearance of a large hot-bed. When fermentation commences, a thermometer should occasionally be introduced into a few of the pots, in order to ascertain the temperature within, which should never exceed 60° Fahr. The depth of the covering of litter, therefore, is to be increased or diminished, according to the state of the fermentation, and partly according to the severity of the season. The vegetation of the included plants is speedily promoted; so that, in the space of a month, the most forward shoots will probably be ready for cutting. The shoots thus produced, being completely excluded from the action of light, are most effectually etiolated, and exceedingly tender and crisp. The advantage of the moveable lids must now be evident: the state of the plants or stools can be examined, and such shoots as are ready can be gathered, without materially disturbing the litter or dissipating the heat. This simple mode of forcing sea-cale has every where superseded the practice of planting it on hot-beds under glass-frames, formerly recommended by Abercromby and other writers on horticulture. This vegetable, it may be remarked, in one respect forms an exception to all others: it is of better quality, when forced in the midst of winter, than when produced naturally in the spring season.

By the modes of culture which have now been described, sea-cale shoots can readily be furnished fresh for the table, from the middle of November till the middle of May.

Rhubarb Stalks.

These are now so much in demand for the making of tarts, that they have become a leading article of trade with the green grocers of London and Edinburgh. The practice of using them seems peculiar to this country; at least it is unknown to the French, the Dutch, or the Germans. The stalks at present sent to market are evidently of finer quality than in former years. By the mode of culture practised, especially the employment of young seedling plants only, the frequent removal of the leaves, and preventing the plant from flowering, the leaf-stalks are rendered more tender than those of plants which have been long established in a garden. Indeed, some of the varieties which have been raised from seed, especially by Messrs Peacock of Edinburgh, have leaf-stalks of a more succulent nature than usual. These appear to be intermediate varieties; and have been raised from seeds yielded by plants of Rheum rhaboticum, growing close by R. hybridum, compactum, and Sibiricum,—the leaf-stalks of which species are used indiscriminately. Such succulent stalks, when peeled, cut down and baked into tarts, have all the appearance of apples, and are by many people preferred to them. In the open ground the stalks are produced from April till midsummer. The progress of vegetation may be hastened during the month of March, by throwing over the plants some loose haulm, care being taken not to injure the shoots, which at that season are very brittle.

Rhubarb may be forced, much in the manner above described for sea-cale; and the leaf-stalks Rhubarb. are thus not only rendered tender, but, being at the same time blanched, become of a fine light colour, and have less of the peculiar flavour of the plant, which is an advantage. The smaller species, such as R. crispum undulatum, are best for this purpose, being most easily confined within the covers. In the third volume of the Transactions of the London Horticultural Society, a mode of forcing the rows of rhubarb, by means of an open frame of wood-work, surrounded with stable litter, is described. Stakes between three and four feet long are driven into the ground opposite to each other, on each side of the row of plants, making the included bed or row two feet wide. The stakes are contracted at top, or made to slope inwards, by means of connecting cross pieces, fifteen inches long. Two or three lath-sparrs are nailed horizontally, along the side stakes, in order to keep the litter from falling in upon the plants. The lining of dung should not be less than eighteen inches thick; the longest litter should be reserved for the top, so as to be easily removed when the state of the interior is to be examined. This plan is well adapted for forcing and blanching the larger species of rhubarb, which could not be confined within sea-cale covers.

Mr Knight (who gives his attention equally to humble details of practical utility, and to philosophical speculations connected with horticulture, and whose name will often fall to be mentioned in this article) has described a method of forcing rhubarb by planting in pots. In the beginning of winter a number of roots of rhubarb are dug up, and placed in some large and deep pots, each pot being made to receive as many as it will contain. Some fine sandy loam is then washed in, so as closely to fill the interstices between the roots, the tops of which are so placed as to be level with each other, and about an inch below the surface of the mould in the pots. The pots are placed in any kind of hot-house; and other pots of the same size are inverted over them. If water be freely supplied, vegetation proceeds very rapidly: three successive crops of leaf-stalks may generally be obtained. The shaded spaces of vineries or peach-houses, which are generally wholly unoccupied, are exceedingly well suited for forcing rhubarb in this manner. Brussels Sprouts.

This culinary vegetable, which is allied to the savoy, originated long ago in the Low Countries, and, as may be inferred from the name, is much cultivated in the neighbourhood of Brussels, where it is called Chou à jets. From the axillae of the stem-leaves proceed small rosettes or sprouts, which resemble savoy cabbages in miniature; these by degrees push off and supplant the main leaves. The sprouts are very delicate when boiled, and justly held in estimation for the table. The culture is nearly the same as that of other coleworts. The seed should be sown in the spring months, and the seedlings planted out before midsummer, during showery weather. The plants grow tall, often three feet, and the sprouts closely surround the stem, the whole forming a narrow pyramid; they may therefore be placed more near together than others of the cabbage tribe, or they may be planted between rows of winter spinach or other low growing crops. In October the plants should have additional earth drawn towards their roots, to firm them, and save them from being destroyed by the frost. The earliest sprouts become fit for use in November; and, if the weather be mild, they continue good, or even improving in quality, till the month of March following. Two or three plants of the most genuine character, with the rosettes small and closely set on the stem, should be allowed to run to flower, in order to secure a supply of true seed. From February till April, Brussels sprouts are now very common in the London market; but they are only beginning to be cultivated in the sale-gardens at Edinburgh.

Mr Van Mons of Brussels mentions (Lond. Hort. Trans. Vol. III.) that, by successive sowings, the sprouts are there obtained for the greater part of the year. The tops of the plants are commonly cut off a fortnight before beginning to gather the sprouts; this, it is thought, promotes the production of rosettes. The sprouts are preferred when small or young; if they be more than half an inch in diameter, they are thought too large. In the spring, when the plants have a tendency to run to flower, their growth is checked, by lifting them and replanting them, in a slanting direction, in a cool shady situation.

Cape Broccoli.

This is an early purple variety, which was introduced a few years ago, from the Cape of Good Hope, according to some, and from Italy according to others. It is a fine kind, being of a delicious flavour when dressed; but on account of the plant being very apt to start into flower, its cultivation has in many places been neglected. When the crops are properly managed, however, this tendency can be overruled. Two crops should be sown; the first in the middle of April; the next in the middle of May. The first sowing may be made on any border of light soil, scattering the seed very sparingly. In about a month the plants may be transplanted, directly from the seed-bed, into a quarter consisting of sandy loam, well enriched with rotten dung. They should not stand nearer than two feet apart every way. Frequent hoeing is proper, and the earth should be drawn to the stem, as in the case of common broccoli. The greater part of the second crop should be planted in pots likewise directly from the seed-bed. These pots are to be sunk in the open ground till the broccoli heads be formed. In the end of November, the pots are to be raised and placed under a glass-frame; and in this way very fine broccoli may be produced in the severest weather of winter. In August, a small sowing should be made in a frame, by which means the plants are somewhat forwarded, without being rendered more tender; these are planted out about the middle of October, three or four together, and protected by hand-glasses during winter. The principal use of this last sowing is to secure the possession of a few good plants in the spring, which may furnish a supply of proper seed.

Knight's Marrow Pea.

It was on the pea, it may be observed, that Mr Mr Knight's Knight first made his experiments, many years ago, New Pea. on the fecundation of one pistillum, by pollen taken from different varieties of blossom, white and grey. In the course of these experiments, he obtained the new pea now to be described. The plant is of luxuriant growth, generally rising to the height of eight or ten feet: in exposed situations it is apt to be injured by the winds; but in sheltered places, and with the aid of tall stakes, it proves extremely productive. The blossoms are white and of large size; and both the legumes (or pods) and the seeds (or peas) are large. The peas are of a cream colour; immediately as they begin to dry, they shrivel or contract in some degree; and, from this circumstance, the name of Wrinkled Pea is often used, particularly among seedsmen. The flavour of the peas, when boiled, is peculiarly rich, surpassing that of any of the other marrow peas: they have been found to abound more in saccharine matter than any others. It is a late pea, and should not be sown before April or May. It makes an excellent principal crop; and it may be added, that it retains its flavour in the autumn better than any other, and should, therefore, be preferred for the latest sowings.

The mode in which Mr Knight manages his autumnal crops of this pea may here be mentioned, because of Peas. it generally has the effect of keeping them free from the attack of mildew. The seed for these crops is sown, at intervals of ten days, from the beginning to the end of June. The ground is dug over in the usual way, and the spaces to be occupied by the future rows of peas are well soaked with water. The mould upon each side is then collected, so as to form ridges seven or eight inches above the previous level of the ground, and these ridges are well watered. The seeds are now sown, in single rows, along the tops of the ridges. The plants grow vigorously, owing to the depth of soil and abundant moisture. If dry weather at any time set in, water is supplied profusely once a week. In this way the plants continue green and vigorous, resisting mildew, and not yielding till subdued by frost. Mouse-Peas.

Under this name, a species of Lathyrus (L. tuberosus) is by some persons cultivated for the sake of the tuberous roots, which being perhaps two inches long, and having a fibre at one extremity, may easily be fancied to resemble mice. When the tubers are of the size mentioned, they are considered fit for use. They are cleared, and, being firm and hard, boiled for a long time, two hours or more, till a fork will pass through them: they are then dried, and slightly roasted; when they are served up in a cloth, in the manner of chestnuts. They are merely calculated for the dessert, and in Holland and Flanders they are not uncommonly used for that purpose.

Mr Dickson, of Croydon, has described the most approved mode of cultivation. (Lond. Hort. Trans. Vol. II.) He recommends the forming of an appropriate border for the plant, inclosed with brickwork, twenty inches deep, and also paved with bricks in the bottom. This bed is filled with a light but rich soil. In this way the roots are restrained from penetrating deep, which they would otherwise do; and the formation of tubers is at the same time promoted. The plant is easily propagated by the tubers, which should be placed six inches apart, and three inches below the surface. The bed should not be disturbed till the second year; after which it will continue productive for a long time, if dug in regular course from one end, leaving the smaller tubers to produce a succession of plants, and adding some good rich soil every year.

Onions.

The cultivation of the onion has been greatly improved by the practice of transplanting. This mode has been recommended in England by Mr Knight, and in Scotland by Mr Brown at Perth, and Mr Macdonald at Dalkeith.

Mr Knight's plan consists in sowing the seed, preferring the variety called White Portugal Onion, at the usual spring season, thick under the shade of a tree, and in poor soil. In the autumn the bulbs are small, scarcely exceeding in size the dimensions of large peas, but of firm texture. They are taken from the ground and preserved till the succeeding spring, when they are planted at equal distances from each other, perhaps six inches in every direction. The plants thus produced differ in no respects from those raised immediately from seed, but in possessing greater strength and vigour, owing to the quantity of previously generated sap being greater in the bulb than in the seed. In this way, two of our short and variable summers produce the same effect as one long and bright summer in Spain or Portugal, and bulbs are procured equal in size and flavour to those that are imported.

Mr Brown's plan, which he has occasionally practised with a part of his own crop for twenty years past, is nearly the same as Mr Knight's, only he does not sow under the shade of trees, with the view of getting small bulbs; he merely collects, from the ordinary onion crop, all the small bulbs, from the size of a pea to that of a hazel-nut (which would otherwise be thrown away as refuse); and having kept these over winter, they are planted in the spring. If the sown beds at any time fail, he can always trust, he finds, to the transplanted rows forming a reserve.

Mr Macdonald confines his operations to one summer. He sows in February, sometimes on a slight hot-bed, sometimes merely under a glass-frame. Between the beginning of April and the middle of the same month, according to the state of the weather, he transplants the young seedlings, in rows about eight inches asunder, and at the distance of four or five inches from each other in the row. Immediately previously to planting, the roots of the seedlings are dipped in a puddle prepared with one part of soot to three parts of earth. The crop being in regular rows, weeds can be destroyed with the hoe in place of the hand, and the bulbs thus enjoy the great and well known advantage of having the surface-earth frequently stirred. Onions of large size are thus produced, equal in firmness or flavour to foreign ones. It is found by experience that the transplanted onions remain free from wire-worm or rot, while those left in the original seed-bed are frequently much injured by both. The beds destined for these transplanted onions are deeply delved over in the beginning of April, and many larvae may probably thus be destroyed; and the plants growing with superior vigour, in consequence of the repeated hoeings, must be better able to resist the attacks of insects. Possibly the soot-puddle may also be beneficial, by tending to repel the larvae till the bulbs be too strong to be attacked. Mr Macdonald finds the Strasburg or Deptford onion answer equally well for transplanting as the Portugal or Reading onion.

Potatoes.

The varieties of the potato cultivated in Britain, having been chiefly derived from Ireland, where the plant is nearly secure from frost from the middle of April till the end of November, the want of new and more hardy varieties has long been felt; and the Horticultural Societies both of London and Edinburgh have offered premiums for the production of such varieties. A hardy potato is, however, still a desideratum.

Various new kinds, some of them possessing desirable properties, have indeed, of late years, been raised by cultivators in different parts of the country; but to particularize these seems unnecessary. It may, however, be remarked, that while the Ash-leaved and American Earlies are the kinds with which the Edinburgh market is principally supplied in the months of July and August, a superior early variety abounds even in the neighbouring town of Perth. This is called the Royal Dwarf. The plant Early Royal is distinguished by its broad shining leaves, and by the first tubers forming a cluster of three or four immediately at the bottom of the stem. This last circumstance renders it easy to rob the plant of the earliest and largest potatoes, without disturbing the roots, leaving it to produce a sufficient crop of secondary tubers for seed-stock. The royal dwarf is a dry potato, or rather mealy than waxy; but this is a quality which recommends it to many persons. It is generally fit for use a fortnight earlier than the ash-leaved or the American early. It may be re- marked, that the most desirable early varieties are such as do not show a disposition to send forth flowers; that portion of the substance and vigour of the plant which would go to the formation of flowers, being diverted to the production of tubers.

A very important fact in the cultivation of potatoes was observed, about the year 1806, by the late Mr Thomas Dickson of Edinburgh, viz. that the most healthy and most productive plants were to be obtained by employing as seed-stock tubers which had not been thoroughly ripened, or even by planting only the wet or least ripened ends of long-shaped potatoes. Mr Knight has likewise clearly shown the advantage of using, as seed-stock, potatoes which have grown late in the preceding year, or have been only imperfectly ripened. It is important to know, that if a valuable kind seem to be exhausted or to have lost its good qualities, it may be restored merely by planting the tubers late in the summer, and preserving the produce of this late-planting for seed-stock.

The forcing of early potatoes on hot-beds has long been practised; but it is attended with considerable trouble and expense. Small supplies of young potatoes are now commonly produced, during winter, in boxes placed in the mushroom-house, in the shade at the back of a hot-house, or in a common cellar, if beyond the reach of frost. In October, old potatoes are placed in layers in the boxes, alternating with a mixture of tree leaves, sand, and light mould, until they be full. Vegetation soon proceeds; and there being no opportunity for the unfolding of stems and leaves, the energies of the plants are expended in the production of young tubers. Before mid-winter, these often attain the usual size and appearance of early potatoes; but they are much inferior, being of a watery taste, and having little or no flavour.

It is much to be wished that we should be acquainted with improved modes of storing the principal autumnal crop, so as to preserve the quality unaltered till the following summer. The Reverend Dr Dow, of Kirkpatrick-Irongray, has devised a mode which certainly merits attention. In the autumn, the potatoes are put into small pits, holding about two bolls each. These pits are formed under the shade of a tree, or on the north side of a high wall; and they are covered with straw and earth, according to the usual mode of pitting potatoes. In the end of April, or beginning of May of the following year, the potatoes are examined; all buds are rubbed off, and such as show any tendency to spoil are thrown out. The pits being cleaned out, are nearly filled with water; when this has been absorbed, the potatoes are returned into them; at the same time, every quantity is watered as it is laid in, and the whole covered with earth, as before. The pits must, in this way, long remain cool. The abundant supply of moisture is, however, contrary to established prejudices as to the mode of keeping potatoes; and on this account, many have probably been deterred from adopting the Doctor's plan. But, in this way, we are assured, the potatoes are kept not only plump and unaltered in taste, but the dry kinds, after being seven months in the pits, come out unimpaired, and appear on the table as mealy as ever.

Turnips.

Nothing new occurs in regard to the culture of turnips, the turnip, unless, perhaps, the practice of sprinkling powdered quicklime over the young plants while in seed-leaf, in order to check the ravages of a little beetle called the turnip-fly. The variety called Stone-turnip is still very much cultivated for the London market. But the Aberdeen Yellow Turnip is preferred, in many places, for use at the table during the winter months. It is hardy, and remains firm and good till the spring.—A very beautiful yellow variety has of late been cultivated, under the name of Maltese Turnip. It is of a round shape, and has such a fine golden colour and so very smooth a skin, that it resembles some foreign fruit. It is excellent for the table; but, if intended for winter use, it must be carefully packed in sand, being otherwise apt to shrivel and decay.—The Swedish Turnip, or Ruta Baga, is now preferred by many persons for the winter supply, on account of its rich flavour and agreeable sweetness. It may either be stored among sand, in a cellar, or, being extremely hardy, it may remain in the ground till wanted.—The Navew, or Navet of the French, is a distinct species, a variety of our native Brassica Napus. The cultivation of the French turnip was promoted in this country during the late war, owing to the numerous French emigrants creating a demand for it. The cultivation is similar to that of ordinary turnips. The root, which is oblong, or carrot-shaped, is of a much higher flavour than any of the common turnips. It is put whole into soups, after being merely scraped, not peeled.

Turnip-rooted Cabbage.

Of the turnip-rooted cabbage, or kohl-rübe, there Kohl-rübe are two varieties, one swelling above ground, the other in it. Both are occasionally used for the table, and, while in a young state, are equal in flavour to the Swedish turnip. There is nothing particular in the culture, unless that, in the case of the first-mentioned variety, the earth should not be drawn so high as to cover the globular part of the stem, or the part used. The seed may be sown in the beginning of June, and the seedlings transplanted in July; they are thus fit for use at the approach of winter; and they may either be stored like turnip, or, being quite hardy, they may be left in the ground till required.

Succory

is, like the navew, a plant indigenous to our island Succory, (Cichorium Intybus), and we also owe its cultivation to the foreign refugees during the war. It is still but little attended to, probably less than it deserves. It is much esteemed by the French as a winter salad; and, when blanched, is known under the name of Barbe du Capucin. When intended for winter Leaves use, the seed is sown in June or July, commonly in Blanched as drills; and the plants are thinned out to four inches Salad. apart. If the first set of leaves grow very strong, owing to wet weather, they are cut off, perhaps in the middle of August, about an inch from the ground, so as to promote the production of new leaves, and check any tendency to the formation of flower stems. In the end of September, or beginning of October, the plants are raised from the border; all the large leaves are cut off, taking care not to injure the centre of the plant; the roots are also shortened. They are then planted in boxes filled with rich mould, pretty close together. These boxes are set in any sheltered situation, and occasionally watered, if the weather be dry. When frost comes on, they are protected by a covering of any kind of haulm. As the salad is wanted, the boxes are successively removed into some place having a moderately increased temperature, equal perhaps to 55°, but not exceeding 60° Fahr. The less light they are subjected to, the blanching is of course the more easily accomplished. The mushroom-house, a corner of the green-house, or a cellar off the kitchen, will answer the purpose. Each box affords two crops of the blanched leaves, a short interval being allowed for the growth of the second crop. The leaves are reckoned fit for cutting when they are about six inches long. A more simple and easy, but perhaps less neat and less productive, mode may be mentioned. The plants may be taken from the open border at the approach of winter, with balls of earth attached to them, placed in boxes, and the interstices between the balls filled with sand. If the green leaves be cut over, and the boxes be placed in a darkened cellar, or other similar situation, a crop of blanched salad will soon be produced.

When colonial produce was excluded from most of the continental markets, the roots of succory were resorted to as a substitute for coffee-beans, and many still continue to use a mixture of succory and coffee, in preference to the simple infusion of the latter. The roots are taken up when of the size of small parsnips; they are cut into little pieces, of nearly equal size; these are carefully dried, generally in an oven, so as to preserve their plumpness and avoid shrivelling; and they are afterwards reduced to a powder in the manner of coffee-beans, as needed for use. The succory root is thought to communicate to the infusion the power of acting as a gentle diuretic.

American Cress,

although its name would lead us to expect a distant origin, is a plant indigenous to England, the Erysimum praecox of the Flora Britannica. It resembles the common winter-cress, E. barbarea, but is smaller; and it is only a biennial, while the former is a perennial plant. The leaves of the American cress have a pleasant warm taste; while those of the common winter-cress are rather nauseous. It has of late years been very generally cultivated as a green salad plant. It may be sown either as broadcast, or thinly in drills a foot asunder, on any light soil. Two or three successive sowings may be made during the season, in order to have young plants; but it may be noticed, that when the outer leaves are regularly gathered, new ones are produced in succession. A late sowing should be made in August or September, on some sheltered border; the plants stand the winter without injury, and afford leaves fit for use in February or March.

Melons.

The melon-ground is generally regarded as an appendage of the Kitchen Garden, and has been treated of in the Encyclopaedia (article Gardening, Part III.) under that head. To the ample instructions there given for the cultivation of the melon, little remains to be added, excepting a caution, founded on the observations of Mr Knight, against removing any leaves for which room can possibly be found. This is the more necessary, that many gardeners of the old school are very apt to think that, of preserving in thinning out the leaves, they are doing service, by admitting sun and air to the fruit, while they are probably inflicting a positive injury. The success of the fruit depends very much on the plant possessing a luxuriant and healthy foliage, having the upper surfaces regularly presented to the light, and remaining as much as possible undisturbed in that position. Pegs are therefore to be freely employed, not only with the view of retaining the shoots in their place, but of keeping the leaves steady and upright; and when water is necessary, it is to be introduced without touching the leaves.

Two uncommon varieties of the melon, introduced of late years, may be shortly noticed; the Salonica and the Valencia.—The Salonica Melon is nearly of Salonica Melon. a spherical shape, and without depressions on its surface; its colour approaches that of gold; its pulp is pure white, of the consistence of that of the water-melon, and very saccharine. The fruit should remain on the plant till it be completely matured; for it improves in flavour and sweetness till it become soft and be ready to decay.—The Valencia Melon is produced plentifully in the countries bordering on the Mediterranean. It is remarkable for the property of keeping for many weeks; insomuch that it has sometimes been imported into London from Spain. In this country it is raised in the manner of other melons. The fruit gathered, when nearly ripe, and suspended in a dry airy room, will keep till January or February. Hence it is often called the Winter Melon. It is oval-shaped, and somewhat pointed at the ends; the skin thin, and of a dark green colour; the pulp whitish, firm, saccharine, and juicy: though the flavour is not rich, it is pleasant to the taste.

Succada.

A small green gourd has for some years past been cultivated in the neighbourhood of London, under the name of Succada or Vegetable Marrow. It may be raised in the spring on a common melon or cucumber hot-bed; and in June transplanted to the open border, in a good aspect, and trained to a small temporary trellis. When the fruit is of the size of a hen's egg, it is accounted fit for use. It is dressed in salt and water, squeezed, and served up in slices on a toast.

Mushrooms.

The usual mode of raising mushrooms, as well as of preparing the spawn, has already been described (Encyclopædia, Vol. IX. p. 443). But what is called Oldaker's method may deserve to be particularized. In forming the compost, he procures fresh short dung, from a stable, or from the path of a horse-mill. The dung must neither have been exposed to wetness, nor subjected to fermentation. There is added about a fifth part of sheep's droppings, or of the cleanings of a cow-house, or of a mixture of both. The whole ingredients are to be thoroughly mixed and incorporated. The beds, if they may be so called, are formed in coarse wooden boxes or drawers. A stratum of the prepared mixture about three inches thick, being deposited in the box, is beat together with a flat wooden mallet. Another layer is added, and beat together as before; and this is repeated till the beds be rather more than half a foot thick, and very compact. The boxes are then placed in the mushroom house, or in any out-house, where a slight increase of temperature can be commanded. A degree of fermentation generally soon takes place in the mass; but if heat be not soon perceptible, another layer must still be added, till sufficient action be excited. When the beds are milk warm (or between 80° and 90° Fahr.), some holes are dibbled in the mass, about nine inches apart, for receiving the mushroom spawn, which, it is to be presumed, has been previously prepared. The holes are left open for some time; and when the heat is on the decline, but before it be quite gone, a piece of spawn is thrust into each opening, and the holes are closed with a little of the compost. A week afterwards, the beds are covered with a coating, an inch and a half thick, of rich mould, mixed with about a fifth part of horse droppings. This is beat down with the back of a spade, and the bed may then be accounted ready for producing. The apartment is now kept as nearly and equally at 55° Fahr. as circumstances will allow. When the boxes become very dry, it is occasionally found necessary to sprinkle over them a little soft water, but this must be done sparingly, and with great circumspection. The more that free air can be admitted, the flavour of the mushrooms is found to be the better; but the exclusion of frost is indispensable. If a number of boxes or drawers be at first prepared, a few only at a time may be covered with mould, and brought into bearing; the rest being covered and cropped in succession, as mushrooms may happen to be in demand. In this way, they may be procured at every season of the year.

Preserving of Cauliflower during Winter.

It is found that this vegetable may be kept in perfection over winter by very simple means. Cauliflowers which have been planted out in July, will be nearly ready for use in October. Towards the end of that month, the most compact and best shaped are selected, and lifted carefully with the spade, keeping a ball of earth attached to the roots. Where there are peach-houses or vineries, the plants are arranged in the borders of these, closely together, but without touching. Some of the large outside leaves are removed, in order that the plants may occupy less room, and at the same time any points of leaves that immediately overhang the flower (or catable part) are cut off. Such houses, however, are generally kept not only without fire-heat, but as cold as possible, during the first part of winter; in time of frost, therefore, it is necessary to cover the cauliflower plants with mats and straw. Another mode consists in placing the cauliflower plants, raised with balls of earth as before, in hot-bed frames, as closely together as possible, without touching. In mild dry weather, the glass-frames are drawn off; but they are kept carefully closed during rain; and when severe frost occurs, they are thickly covered with mats. If the plants be occasionally cleared of decayed leaves, they will continue, in this way, in excellent state for several months, instead of becoming yellow and ill-flavoured, as they generally do when placed in sheds or cellars, where air and light cannot occasionally be given.

Keeping of Vegetables, &c. in the Ice-house.

The Ice-house is generally under the care of the Ice-house gardener; and where it is placed near the garden, it is found useful for several subsidiary purposes, and particularly for preserving esculent roots, and likewise celery, during winter, in recesses contrived for the purpose. Where parsnips and beet-roots are left in the ground over winter, they must be lifted at the approach of spring, as they become tough and woody whenever there is a tendency to form a flower-stalk. These roots may, therefore, at this season, be placed in the ice-house, and preserved there for a considerable time in excellent order. The ice-house is equally useful in this respect during the summer season: in hot weather, various kinds of vegetables, for instance green peas and kidney beans, can be kept fresh in it for several days.—In order to avoid introducing the subject again, another use may here be mentioned: fruits gathered in the morning, which is the most proper time for gathering them, are here kept cool, and with all their freshness and flavour, until required for the dessert in the afternoon. Several ice-houses, excellently adapted not only for the main purpose, but for these secondary views, which nowise interfere with the other, have lately been constructed in the neighbourhood of Edinburgh, under the directions of Mr Hay, planner of gardens, particularly at Dalmeny Park and Dundas Castle. These ice-houses have double walls, a passage being left between the outer and inner. In the thick wall immediately inclosing the ice are four recesses, with stone shelves for receiving the vegetables or fruits. In the outer wall the same object is provided for. The roof, it may be added, is arched with stone, and has a hole at the top for introducing the ice. The passage between the two walls is likewise arched, and has two or three small grated apertures, which may be closed with fitted stones, or opened for the purpose of admitting light and air when wanted.

FRUIT GARDEN.

During the last thirty years the desire for fruit has greatly increased among the inhabitants of this country, and the attention paid to its production has advanced in proportion. The general diffusion of this taste has created such a demand in the ne- tropolis and principal towns, that not only are professional cultivators enabled to lay out considerable capitals with advantage in the raising of exotic fruits, but great encouragement is thus given to private gentlemen to improve and enlarge their gardens, vineries, and peach-houses; because a ready and lucrative market is open for the superfluous produce at any time, and for the whole produce of the garden, when the proprietor and his family happen to be from home. We shall first advert to any changes or improvements in the general management of the garden, or of the different kinds of hot-houses connected with it; and shall next take particular notice of the new fruits, or new varieties, which have been lately introduced, or have lately risen into notice.

Fruit-Trees in the Nursery.

Some kinds of fruit-trees, particularly the Mulberry and Walnut, are so slow in their progress to a bearing state, that the planter of the trees seldom sees their fruit. Mr Knight has ascertained that, if the cions be taken from prolific branches of bearing trees, the young trees become productive in a very few years. Indeed, if the stocks be planted in pots, and grafted by approach, they afford fruit in three years after the operation. Young trees thus grafted with cions from the bearing wood of adult trees, are not yet to be found in the public nurseries; even the most eminent of our nurserymen not possessing a collection of bearing trees for this purpose.

In regard to the training of young trees, especially of the peach and pear kinds, notice may be taken of an excellent and simple mode for which we are indebted to Mr Knight. His plants are headed down as usual, a year after being grafted; two shoots only are allowed to each stem, and these are trained to an elevation of about 5°. It is a well known fact in horticulture, that a branch trained upright grows much more luxuriantly than one confined to a horizontal position. Advantage is here taken of this law of vegetation, and in order to procure the shoots to be of equal length, the stronger is depressed and the weaker elevated. All lateral shoots are carefully removed. Next season as many branches are encouraged as can be laid in without overshadowing each other; and if care be taken in the spring to select the strongest and earliest buds near the termination of the year-old branches, to be trained lowest, and the weakest and latest buds near the base of the branches to be trained inclining upwards, the result is, that, at the end of the season, each annual shoot comes to be nearly of equal vigour. In the following winter, one half of the shoots are shortened, and the other half left at full length, one shoot being left and the other cut alternately. In the third year, if the subject be a peach-tree, the central part will consist of bearing wood. The size and general health, and equality of vigour in every part, of young trees trained according to these rules, appear to evince a very regular distribution of the sap; and the rules are simple, and might easily be attended to.

VOL. IV, PART II.

Wall Training.

As the trees above described advance, they naturally fall to be trained in what is called the fan mode, or according to various modifications of this. Where the garden-walls exceed seven feet in height, this is the mode now preferred by the best practical gardeners; for in this way a tree can much sooner be made to fill the space of wall allotted to it, and the loss of a branch can most easily be supplied at any time. The fan mode is particularly well adapted for such kinds of fruit-trees as do not abound in superfluous wood, or extend their branches to a great length, as the peach, nectarine, apricot, and cherry trees. For walls under seven feet in height, the horizontal method of training is still preferred, as in this way the wall can be more completely filled, although not in so short a space of time. In this mode, which was first strongly recommended by Hitt in his excellent Treatise on Fruit-Trees, a principal stem is trained upright, and branches are led from it horizontally on either side. Many kinds of pear-trees, and also apple-trees, are very productive when trained in this horizontal manner.

In both modes of training, and with all kinds of trees, it has been found very advantageous to have the extreme branches bent downwards. By this means a check seems to be given to the growth of the wood of the tree, and a tendency to yield fruit is promoted. Besides, it is evident, that, in the flexure of the extremities of wall-trees, the natural mode of growth is imitated.

Connected with this subject is the recent practice of turning the extreme branches of fruit-trees from one side of a wall to the other. The late Sir Joseph Banks having a Gansel's bergamot pear-tree on a north aspect, where the fruit did not succeed, caused some branches be turned over to the south side, and trained downwards. There they not only produced fine fruit, but abundance of it. The roots of the May-duke cherry, and some others, require to be in a cool soil. On the north side of a wall, therefore, such trees thrive best; and it has been found, that if their extreme branches be turned over the wall, and trained downwards on the south side, they are not only brought into plentiful bearing, but yield their fruit more early in the season.

Before leaving wall-trees, we may here notice, that, for protecting the blossom of peaches and nectarines from the effects of hoar-frosts and cold dews, nets made of coarse woollen yarn or carpet worsted have, in some parts of Scotland, been very advantageously employed. When such nets are worked in the loom, they can be afforded at a very cheap rate. They are woven pretty close, the meshes not being larger than to admit the point of the finger. Worsted nets are better than any other, on account of the bristliness of the material and its tendency to contract. Screens covered with white paper have likewise been employed with good effect. Where such screens are made to project sufficiently from the wall, and are applied in the evenings, they will be found very effectual in preventing the radiation of heat from the earth in the cold and clear nights which often follow warm days in May and June; and not the setting fruit only, but tender plants, such as love-apple, may thus be protected. The importance of this remark will be evident to all who have attended to the doctrines of Wells and Professor Leslie on the subject.

Standard fruit-trees, particularly pears, are now frequently trained in a pyramidal form, or what the French term en quenouille. This is effected by preserving only an upright leader, and cutting in the lateral branches every year. Trees managed in this manner occupy much less room, and throw much less shade, than when allowed to spread their branches at will. If thought proper, they may likewise be planted very near together without injury; six or at most eight feet being a sufficient space between such trees. In general these pyramidal trees are very productive. They are not well calculated, however, for places subject to high winds, but rather require a sheltered situation. In appearance they are stiffly symmetrical, and the lover of the picturesque in gardening, would greatly prefer the natural spreading of the tree.

Apple-trees are now very generally trained en buisson, or as dwarfish standards, and in this form they can be scattered along the borders of the garden without producing inconvenience.

Particular varieties of apple are observed to succeed in certain soils and situations better than in others: it is the business of the cultivator to take notice of these, and to multiply them by grafting. At the garden at Dalkeith belonging to the Duke of Buccleuch, where the soil is shallow and the subsoil unfavourable, great crops of apples are yearly produced, merely in consequence of planting shallow and of frequent grafting. Mr Macdonald, the excellent gardener officiating there, annually inserts on his numerous trees not fewer than from 2000 to 3000 grafts, generally three or four sorts on each tree. The grafts are chiefly of such kinds as experience has taught him to prove generally successful at Dalkeith garden.

Decortication.

When the outer bark of fruit-trees, especially of the apple kind, becomes rough and cracked, so as to admit minute insects to deposit their ova under it, it has for a long time been the practice to remove it entirely, and to cleanse the trunk and principal branches with some kind of wash. This partial decortication and cleansing, it was observed, not only produced a healthy foliage, but had an evident effect in promoting the fruitfulness of the trees, or in causing the conversion of leaf-buds into flower-buds. Of late years, Mr Lyon of Edinburgh, founding wholly on his own experience, has particularly called the attention of the public to the practice of decortication; and he has invented several simple instruments for facilitating the removal of the bark. He has carried the practice much farther than his predecessors, who, as already noticed, removed the bark only when it was somewhat diseased, and only from the trunk and larger branches. Mr Lyon recommends the stripping even of young trees, and of the new shoots of full grown trees, however healthy the bark may be. Even where the bark of a tree is healthy, a partial removal of it (as in the practice called ringing, presently to be noticed) may prove beneficial, in causing the production of fruit-buds; but it is evident that a useful practice may be pushed too far.

The decortication of vines has likewise been re-vived, and has been strongly recommended of late, in a pamphlet by Sir John Sinclair, who founds particularly on the experience of Mr King, an active and industrious fruit-gardener at Teddington in Middlesex. The operation is performed in the beginning of winter, with a common knife; for the outer bark may, at that season, be easily separated from the inner concentric layer, without hurting the latter. Not only are the plants thus treated freed from numerous small insects, which never fail to make a lodgement in the crevices of the rough parenchymatous bark, but they are observed to make stronger shoots, and the quantity of grapes is said to be increased, and their quality improved.

Analogous to this is the practice of ringing of the branches of vines, or making a narrow annular incision, and removing a ring of the bark: in this case, both the outer and the inner bark is removed. The consequences of this practice are said to be very beneficial. The same plan of removing a ring of bark, about a quarter of an inch in breadth, and down to the albumum, has been practised on apple and pear trees, by different cultivators, with considerable success; the trees being thereby not only rendered productive, but the quality of the fruit being at the same time apparently improved. The advantage is considered as depending on the obstruction given to the descent of the sap, it being thus more copiously afforded for the supply of the buds. The ring should therefore be made in the spring; and it should be sufficiently wide, that the bark may remain separated for the season in which it is made. None of the stoned fruit-trees are benefited by ringing.

Pruning of Currant Bushes.

An improvement in the management of the currant-tree deserves notice. Mr Macdonald at Dalkeith, whose name has just been mentioned, prunes the bushes at the usual season of midsummer, shortening the year's shoots down to an inch or an inch and a half. Next summer the plants generally show plenty of fruit, and at the same time send out strong shoots. As soon as the berries begin to colour, he cuts off the summer shoots to within five or six inches before the fruit. For the sake of expedition, this operation is commonly performed with the garden shears. Sun and air thus get free access to the fruit, and more of the vigour of the plant is directed to it: in consequence, the berries are found to be not only of higher flavour, but of larger size.

Hot-houses.

All the different kinds of glazed houses employed for the production of the more tender exotic fruits, have in some respects received improvements. But pine-stoves have undergone the greatest change of structure. In place of the lofty wide houses of former times, small low pits are now employed. These are commonly of two sizes; one, called the Succession pit, is rather lower in the roof and of smaller dimension than the other, which is the Fruiting pit. The advantages are considerable: the atmosphere of these last can much more easily be maintained at the requisite temperature; and the plants enjoy the advantage, well known from experience, of being placed near to the glass. In propagating ananas, some of the most successful cultivators use suckers only: these are allowed to remain long on the parent plants, so that when they come to be detached they are of a larger size and more forward growth than is usual. The suckers are planted in pots in September, and placed in beds of tan, in any common hot-house furnished with a furnace and flues. After the plants have fairly made roots, a high temperature is not wanted, and, for the following six months, if frost be carefully excluded, the plants succeed best in a cool house, which may be supposed somewhat to resemble the winter of their native country: pretty late in the spring, they are transferred to the pine-pits. Sometimes this is in reality little else than a large hot-bed having tanners' bark in the centre, and being furnished with exterior linings of stable litter, or some other fermentable material. In other cases the pit has likewise a furnace and flues: in those pits, however, which depend on fermentation alone for artificial heat, the ananas are observed to grow remarkably fast during the summer season. In autumn the plants are again returned to the common hot-house for the winter: in the course of the following season they are brought to fruit in the larger sized pit; and if this be not of sufficient dimensions, as sometimes happens from the spreading of the plants, a few of the most forward are allowed to fruit in their winter quarters. In this way pine-apples, particularly of the variety called the Queen, are produced in two years, instead of three, which were formerly thought necessary.

In the opinion of Mr Knight and of other eminent cultivators, the employment of a bark-bed, or bottom heat of any kind, is wholly unnecessary after the crowns or suckers have pushed their roots. In an ordinary hot-house, the pots may be placed on loose piers of brick, and thus raised near to the glass; a layer of bricks being removed as the plants increase in height. In the summer season, the temperature may depend chiefly on confined solar heat, no air being given till the temperature exceed 95° Fahr. For soil Mr Knight prefers thin green turf chopped small, and pressed close into the pots while damp; a piece of whole turf, with the sward downmost, being laid at the bottom of the pots. The surface, however, is covered with vegetable mould and sandy loam mixed. Mr Knight recommends applying daily to the pots, during the height of summer, water in which pigeons' dung has been steeped till the colour be nearly as dark as that of porter. A little pure water may be sprinkled over the plants, but not till all remains of the former sprinkling have disappeared. As the day gets shorter, less of the pigeons' dung water is given, the plants being then less able to feed on it. During winter the house is kept as nearly as possible at 50° Fahr. Mr Knight prefers pots which are little more than a foot in diameter, and he does not seem to consider repotting as necessary: at least, he regards the shifting from smaller to larger pots as detrimental, the matter which would go to the formation of blossom and fruit being thus diverted to the production of new roots.

It may here be remarked, that for communicating heat to pits or frames, it has been found advantageous, in place of stable litter, to employ the cleanings of a flax-dresser's mill, known under the name of Lint-shows or Flax-pob. This substance ferments very slowly, and the heat is therefore kept up for several months in succession, and very nearly of an equal temperature.

Various improvements and changes in the form and interior arrangements of glazed houses intended for the production of peaches, nectarines, figs, and grapes, have of late been introduced or recommended. These are detailed chiefly in the Transactions of the Horticultural Society of London, and in the Memoirs of the Caledonian Horticultural Society. Mr Knight and Mr Gowen, with Mr Loudon at Bayswater near London, seem to be the principal persons who have attended to these subjects in England; and Mr Hay of Edinburgh, Mr Beattie at Scoon, and Mr Henderson at Brechin, have led the way in Scotland. Among amateur horticulturists, our countryman Sir George Mackenzie has distinguished himself by projecting spherical hot-houses; and modifications of this form have been strongly recommended by Mr Loudon.

A very considerable improvement in the mode of glazing hot-houses may deserve to be more particularly mentioned, because it tends materially to obviate breakage, which, on account of the high duty on glass in this country, is now an important object. It consists chiefly in making the upper and lower edges of the panes segments of a circle, instead of being rectilinear or horizontal; the upper edge being made concave, the lower convex. For a pane eight inches wide, a curvature 8ths of an inch deep in the centre is sufficient. The advantages of this circular form must be evident. The rain which falls, or moisture which collects on the exterior of the glass, gravitates to the centre of the pane, and runs down in a continued line, instead of passing along the sides of the bars, and being partly detained by the capillary attraction of the two surfaces, at the overlapping of the panes. The extent to which one pane overlaps another can, at the same time, therefore, be much lessened; and 1/6th of an inch is found sufficient. This narrowness of the lap, again, prevents breakage from the lodging of moisture, and the sudden expansion produced by freezing during the variable weather of winter. When these circular panes are cut from whole sheets of glass, the expence is scarcely greater than for oblong squares. It is proper that the glass should be very flat or equal; and the kind known by the name of Patent Crown Glass should be preferred. In stoves or hot-houses where a high temperature must be maintained, the laps are putted. In this case, a small central opening is left in the putty, by inserting a slip of wood at first, and withdrawing it when the pane is pressed down to its bearing; by this little aperture the condensed vapour generated within escapes without dropping on the plants. The ingenious Mr Loudon uses very thin sheet lead in place of putty, for closing the laps; he thus avoids all risk of expansion from frost, and the lap can thus be made exceedingly narrow.

Heating of Hot-houses by Steam.

Of all recent improvements, however, in this branch of gardening, the most important is the use of steam for communicating the artificial heat, in place of depending, as formerly, on the passage of smoke and heated air through flues, aided in particular houses, called Stoves, by the slight fermentation of tanners' bark. The principal advantage arising from the use of steam consists in this, that an equable high temperature can thus be maintained for a length of time with much greater ease and certainty. Besides, in steam hot-houses, the plants can scarcely ever be liable to suffer a scorching heat; the air continues pure and untainted, and persons visiting the house are much less apt to be annoyed with the smell of smoke or soot. In districts where coals are scarce and high priced, the saving of fuel is an object; and it has been found that seven bushels of coal go as far in keeping up steam heat, as ten bushels do in maintaining an equal temperature the other way. Further, it is evident that, by merely opening a valve, the house may, at any time, be most effectually steamed, that is, filled with vapour; and the warm moisture thus applied to every part of the plants is observed to contribute remarkably to their health and vigour.

While steam alone may, in new erections, be trusted to for supplying the necessary heat, it fortunately so happens that it may likewise very advantageously be resorted to in aid of the common flues conveying smoke and heated air. A steam-apparatus may be appended to any ordinary hot-house, without incurring any material expence, or occasioning any considerable alteration in structure. A boiler is erected over the usual furnace, the smoke of which passes through the flues as formerly. Metal pipes are laid along the top of the brick-flues. These are rather of copper than of lead, on account of the former expanding less. A square shape is sometimes preferred; and the pipes are set on edge, so that any condensed vapour trickling to the bottom may occupy little room, or present only a small surface, till it make its way back to the boiler, to which a gentle inclination is given. As in the common steam-engine, the boiler is supplied from a cistern above, and is made to regulate itself by a simple contrivance; in the feed-head is a valve, which is opened by the sinking of a float, which descends in proportion as the water is dissipated in steam; and being balanced by a weight, whenever a sufficient quantity of water is admitted, rises again and shuts the valve. A safety-valve is added, loaded according to the strength of the boiler; and there is another valve for admitting atmospheric air, in case of the condensation of the steam causing a vacuum in the boiler. By thus adding a steam-apparatus in aid of the common flues, a higher and much more steady heat can be commanded. Instead of requiring more of the time and attention of the gardener, he will be greatly relieved, and have several additional hours a day which he may wholly devote to other concerns of the garden. If the furnace be duly charged, and the boiler properly prepared, the hot-house may be left with confidence for eight or even ten hours together, the temperature continuing equal for that length of time. Where forcing is practised during the severe weather of winter and early spring, the gardener is thus relieved from much anxiety and night-watching, to which he was formerly subject.

For heating stoves, conservatories, and greenhouses, steam is likewise excellently adapted. The difficulty of maintaining continually a high temperature in a large stove has, no doubt, been one cause of the comparative neglect into which the cultivation of fine tropical plants in England has fallen. By means of steam, this difficulty is most effectually removed: and we may soon expect to see the noble palms, and arborescent ferns of the tropical regions, waving at large in commodious receptacles heated in this manner.

It may here be mentioned, that the cultivation of tender exotics has of late been further rendered easy, by the substitution of a chamber filled with heated air, or with steam, in place of tanners' bark; the procuring of which is often attended with difficulty and expense, and the proper drying of it invariably troublesome. The plants are placed immediately over the steam-chamber, the roof being formed of thin flagstones, like those known by the name of Arbroath pavement. The pots may be sunk in sawings of wood, which remain for a very long time in a clean and unaltered state; and in which insects are not very apt to breed.

For the conservatory and green-house, if the steam be in action from three to nine o'clock P.M., the temperature will be kept constantly within a proper range, in the ordinary winter weather of this country. In time of severe frost, the steam must, of course, be longer applied.

The most extensive and most perfect steam-apparatus for the heating of plant-houses is to be seen at the grounds of Messrs Loddiges, near Hackney, where glazed houses to the extent of almost a thousand feet in length, and forming three sides of a square, are heated solely by steam from a single boiler. The boiler is of an oblong shape, measuring eleven feet by four, and is made of malleable iron. In certain narrow houses, intended for green-house plants, a single steam-pipe is found sufficient. In other houses, of considerable height and breadth, or where a higher temperature is required, as in the palm-house, the steam-flue is made to describe two or three turns. The pipes at Hackney are of iron, of a round shape, and four inches bore. They are flanged and screwed together with bolts and nuts. When they make returns within the house, the joints are formed with iron cement on milboard dipt in white lead.

Where steam is employed for heating the principal suite of hot-houses, it will be found easy, in general, to convey it also to the melon ground; the melon pits or frames must, however, in this case, have their side-walls formed of brick. In places where steam-heat has been applied to the culture of this fruit, the success is said to be remarkably great.

Watering of Green-house and Hot-house Plants.

When large collections of plants are kept in the green-house, hot-house, or conservatory, the watering of them by the hand is a tedious operation. The ingenious Messrs Loddiges, already mentioned, have devised a mode of greatly facilitating this operation. A leaden pipe, of half an inch bore, is conducted horizontally along the upper part of the house, in the space most convenient for the purpose in view. This pipe is everywhere perforated, the holes being so small as only to admit a fine needle. The perforations are so disposed in the pipe as to throw the water in the directions where it is most wanted. In general, the holes are at two inches distance from each other; but, towards the extremity of the pipe, where the pressure of the water is less, they are somewhat closer. The cistern must, of course, be above the level of the pipe. By turning a stop-cock, the water passes along the pipe, and is diffused over the plants, in the manner of a gentle shower of rain.

Production of new Seedling Fruits.

During the last twenty years great attention has been paid to the production of new seedling varieties of the more hardy fruits suited to our climate. For exciting the attention of the public to this important matter, we are particularly indebted to Mr Knight. A very succinct statement of his views on this subject, which have sometimes been strangely misrepresented and even turned into ridicule, may here be proper. In his Treatise on the Apple and Pear, he noticed in a particular manner the fact, that some of the finest cider and perry fruits of the seventeenth century have already become extinct. This fact was undeniable; for daily experience showed, that the golden pippin in England, the grey Leadington and white Hawthorndean in Scotland, and other old apples, were fast wearing out. Mr Knight remarked, that each variety of fruit springs from an individual at first; and that, by means of grafting or budding, the individual only has been extended. Whatever tendency to decay and extinction existed in the individual at first, must, he observed, exist in all the extensions of that individual accomplished by means of buds or grafts. By careful management or fortunate situation, the health and life of a particular individual or original tree may be prolonged; and, in like manner, some buds or grafts, placed on vigorous stocks and nursed in favourable situations, may long survive the other buds or grafts from the same tree, or may long survive the original unengrafted tree. Still, in all of them, there is a progress to extinction; the same inevitable fate awaits them: the only renewal of an individual, the only true reproduction, is by seed.

Mr Knight's doctrine, we may add, seems now to be established as to fruit-trees. It may probably be extended to all trees, and even to all the more perfect tribes of plants; for the sagacious Philip Miller long ago observed, that herbaceous plants propagated by cuttings, became barren in a few years. The importance of acquiring new varieties of our staple fruits from the seed is now, therefore, universally acknowledged; and as a taste for experimenting in this way is prevalent, we may probably do an acceptable service to our readers, in bringing together some of the precautions adopted by the distinguished horticulturist already so often mentioned, and the facilities which have been devised towards success in this interesting branch of gardening.

The seeds to be sown should belong to the finest kinds of fruit, and should be taken from the ripest, largest, and best flavoured specimens of each kind; for although some crab-apples may result from sowing the seeds of the nonpareil or the Newton pippin, yet from the seeds of such excellent varieties, there is a greater chance of procuring an apple somewhat similar in qualities. Mr Knight took uncommon pains in order to procure promising seeds: for example, he prepared stocks of the best kinds of apple capable of being propagated by cuttings, and planted these stocks against a wall in a rich soil; these were next year grafted with the golden pippin. In the course of the following winter, the young trees were raised from the ground, and the roots being shortened, they were replanted in the same spot. By this mode of treatment they were brought into a bearing state at the end of two seasons. Only two apples were suffered to remain on each little tree; these fruit consequently attained a large size and perfect maturity. The seeds of the apples thus procured were sown, in the hopes of procuring seedlings possessed of qualities allied to those of the golden pippin; and if these hopes have not yet been fully realized, the success has been sufficient, at least, to encourage to perseverance in similar modes of experimenting.

It may here be mentioned further, that, with the view of producing a variety uniting the good properties of two known and highly approved kinds, Mr Knight, Mr Macdonald, and some others, have been at the pains to bring the pollen of the one kind in contact with the pistils of the other. To do this with proper effect, requires some nicety and caution. Mr Knight opened the unexpanded blossom of the variety destined to be the female parent of the expected progeny, and with a pair of small-pointed scissors cut away all the stamina while the anthers were yet unripe, taking great care to leave the style and stigmas uninjured. The full blown blossoms of the other variety were afterwards applied. The fruits resulting from such artificial impregnation have been of the most promising character: the seeds of these fruits, again, were sown, with the expectation of procuring improved varieties, and there is every reason to think that the expectation will be realized. Mr Knight has often remarked in the progeny, a strong prevalence of the constitution and habits of the female parent: in this country, therefore, in experimenting on pears, the pollen of the more delicate French kinds, as the crassane, colmar, or chaumontelle, should be dusted upon the flowers (always deprived of stamina) of the muirfowl egg, the grey achan, the green yair, or others that are hardy or of British origin. By these means, it may be hoped that, in the course of another generation, excellent winter pears may be obtained in abundance from our standard trees; for at present we are nearly destitute of hardy winter pears.

Some persons make a practice of sowing great numbers of seeds, taken indiscriminately. Out of some hundreds of such seedlings, a very few only may prove deserving of any notice. In the ordinary course of nature, the lapse of six or perhaps ten years would be required before the fruit could be seen. But in order to form a general estimate of the character of the seedling trees, it is not necessary to wait till they actually produce fruit: even in the first season, such an opinion may, to some extent, be formed, from the shape and texture of the leaves; those which are pointed, thin and smooth, promising little; while those which are blunt or round, thick, and inclined to be downy, promise well. In the second year, these tests are more satisfactory; for the leaves of good kinds improve in the above noticed qualities yearly. Plants whose buds in the new wood are full and prominent, are much to be preferred to those whose buds are small and almost sunk into the bark.

Some means have likewise been devised for hastening the production of the fruit of seedling trees, or shortening the period of probation. The moving of the plants and shortening of their roots have already been mentioned. Mr Williams of Pitmaston, an eminent English horticulturist, has succeeded in promoting their early puberty, by using means to hasten that peculiar organization of the leaf which appears necessary to the formation of blossom-buds. The seeds (of course only of select kinds) are sown in pots, and the growth of the seedling plants is forwarded by the artificial heat of a peach-house or vineyard. They are afterwards planted out in nursery lines. Every winter, all small trifling lateral shoots are removed, leaving the stronger laterals at full length; and such a general disposition of the branches is effected, that the leaves of the upper shoots do not shade those below. Every leaf, by its full exposure to light, is thus rendered an efficient organ, and much sooner becomes capable of forming its first blossom-bud. Those who have even slightly studied vegetable physiology, must be convinced of the great consequence of attending to such apparently minute circumstances.—Another plan resorted to with success, consists in taking cions from the seedling trees, and grafting them on wall-trees in full bearing: in this way, the fruit may be seen in three or four years from the sowing of the seeds. If possesses any promising qualities, such as fine colour, firmness, or flavour, it ought not to be rejected at first on account of acidity or smallness of size: If a seedling be somewhat juicy, it is very promising, for this good quality also increases with its years; and it is remarked, that a fruit having a firm pulp commonly improves with the age of the tree, but that a soft or mealy pulp gets worse. In general it may be remarked, that the fruit has always a tendency to improve in mellowness and in size, as the tree itself becomes stronger and approaches maturity.

Not only have British horticulturists been successfully occupied in producing new varieties at home, but they have at the same time been extremely active in introducing approved kinds raised in other countries. Some of the best new varieties, both native and foreign, of the different fruits usually cultivated, shall now be enumerated.

New Varieties of Fruits.

APPLES.—For the best new varieties of this ex- New varie- centell and useful fruit, we are indebted to Mr ties already Knight. The Downton Pippin has now been known produced, and approved of for a good many years. In exposed or upland situations, it comes of better quality than in low and warm places. The Wormsley Pippin is a large fine fruit, resembling in the consistence and juiciness of its pulp the Newtown pippin. The Yellow Ingesterie Pippin is the produce of one of the hybrid fruits already mentioned, between the golden pippin and the orange pippin: in shape and colour it resembles the former, and it also rivals it in richness and flavour. The Scotch Nonpareil is another of the hybrid productions, for which we are indebted to Mr Macdonald of Dalkeith. It was raised from a fruit produced after dusting the blossoms of the nonpareil with those of the Newtown pippin. By grafting on a wall-tree, he procured the new fruit in the fourth year. Specimens of this fruit have at different times been exhibited at meetings of the Caledonian Horticultural Society, and have always met with great approbation. The Russet Nonpareil was raised at Pitmaston near Worcester, from seed of the nonpareil. The blossom appears to be more hardy than that of the parent variety; the fruit is compressed, of a dull green, much covered with russet; the pulp is of a pleasant consistence, and highly charged with the peculiar aromatic flavour which characterizes the nonpareil. The Martin Nonpareil, raised likewise near Worcester, is regarded as a fine dessert fruit; it is remarkable for keeping in a sound state, not only over winter, but till the following midsummer; for supplying the table, therefore, in the spring months, this variety is valuable. Still another offspring of the nonpareil has been recommended in the Transactions of the London Horticultural Society, called the Braddock Nonpareil; the pulp is sweeter and more melting than that of the nonpareil, richly sugared, and slightly aromatic. The Breedon Pippin is a new English variety which cannot be traced to its original: in shape it is flatly conical, with many plaits or wrinkles around the eye; the skin is of a deep dull yellow; the pulp yellowish, firm, very sweet, with a rich vinous acid. The Lamb Abbey Pearmain is the offspring of an imported Newtown pippin; but it differs very much in shape and general appearance from the Long Island fruit, being of an oval form and somewhat pyramidal; the pulp is yellowish next the skin, and green next the core, very firm, so as to fit for keeping, at the same time juicy, richly sweet, and not without flavour: the tree requires to be trained against a wall or espalier rail, because the branches are so slender that they cannot, in general, support the fruit.

From North America we have, of late years, re- ceived several excellent varieties of apple. The Newtown Pippin from Long Island, already repeatedly named, deserves the first place. It is an excellent dessert apple, allied to the rennets; it keeps well, is in perfection for the table in January, but continues good till March or later. In this country the tree requires a wall with a good aspect. The Spitsenberg Apple is of a fine appearance, and the pulp has somewhat of the pine-apple flavour: the tree requires a sheltered situation and good soil; it succeeds better on a west than an east wall. The American Nonpareil or pomme de grise is a high-flavoured apple, introduced only a few years ago; it ripens very well on a wall having a west aspect. The Canadian Rennet is a large fruit, of a yellow colour, with a tinge of red; it likewise requires a wall in this country. Of varieties brought over from the Continent, we shall only notice the Borsdorfer, which is one of the most highly esteemed throughout Germany. The fruit is round, of a yellow colour, but red next the sun; having a rich flavour, it is suited to the dessert, as well as for all culinary purposes.

PEARS.—For the production of new seedling pears in this country, longer time and more attention are required than in the case of apples. Generally ten or twelve years elapse, before a seedling pear-tree shows blossom-buds. Only two new pears can here be recommended; but several very promising seedlings are known to be in a state of progress both in England and Scotland.—The Wormsley Bergamot has been raised by Mr Knight, from the blossom of the autumn bergamot dusted with the pollen of the St Germain. It is a melting pear, of good flavour. The tree grows freely, and the blossom appears to be hardy.—Williams' Bon Chretien is a large fruit, of a pale green colour; pulp white, very tender, abounding with an agreeably perfumed sweet juice. The tree bears freely, even as a standard; but the fruit comes to greatest perfection on a west wall. It is now a good deal cultivated in the neighbourhood of London.

From America, we have, of late, received an excellent variety, called the Sickle Pear. The fruit is rather small; sometimes of a yellow colour, and red next the sun, at other times altogether of a russet appearance; the pulp is melting, juicy, and of exquisite flavour. The tree is very vigorous and quite hardy.—The public nurseries at Edinburgh have, within these few years, been enriched with grafts of some of the finest seedling pear-trees raised at Brussels by Mr Van Mons, a distinguished cultivator there, and now Professor of Rural Economy at Louvain. None of them have yet produced their fruit in this country; but a Committee of the Caledonian Horticultural Society had an opportunity of tasting several of the fruits at Brussels, and have particularly recommended those called Poire Marie Louise, Poire Napoleon, Marly, Diel, Salisbury, Archduke Charles, and Callebasse.

PEACHES.—In the production of new peaches, Mr Knight again excels. He planted several peach-trees in large pots, and paid every attention to bringing them to a state of high health and vigour; he then applied to the pistil of one good kind the antheræe of another; each tree was allowed to bring to perfection no more than three fruits: from sowing the stones of these some new and improved seedling varieties were looked for, and the expectations have not been disappointed. Two new kinds deserve particular notice; and the situation of Downton, the seat of Mr Knight, being rather high and exposed, it may be presumed, that fruits which are produced there, may probably succeed even in the more northern parts of the island. 1. The Acton Scott Peach. The fruit comes early, and never fails to attain maturity; it is juicy and sweet, with a rich flavour. The tree is a plentiful bearer, and not liable to mildew. This new variety deserves the especial attention of Scottish horticulturists. 2. The Spring Grove Peach is of a bright yellow colour, and red next the sun: it has a firm but not hard pulp, which melts in the mouth, and has a remarkably rich, brisk, and vinous flavour. The fruit never becomes overripe or mealy, but, when quite ripe, is apt to shrivel a little: it is then in the most perfect state for the table. The tree grows slowly, but ripens its wood early in the season. It seems to succeed better on an apricot than a plum stock.

To America we owe Braddick's American peach, figured and recommended in the second volume of the London Horticultural Transactions. It is a large fruit, with a yellow skin, red next the sun; the pulp is yellow, and of high flavour. It is not a hardy kind, nor does the tree produce freely.

NECTARINE.—We know only of one new variety Nectarine, of nectarine, which can at present be recommended for cultivation. This is the Woodhall Nectarine, so called from its having been raised at Woodhall, near Holyton in Scotland, by Mr Walter Henderson, gardener there, well known as a most successful cultivator of the Citrus and Erica tribes. The fruit approaches most nearly to the elruse; but it is more juicy, and perhaps also of a higher flavour; the fruit never fails to come forward to maturity. At present its good qualities are evidently on the increase. The tree grows freely, and has never shown the slightest symptom of mildew; the wood ripens readily in the autumn. The blossom is small, early, and hardy; and ever since the tree came into bearing, about six years ago, it has not once failed to produce an abundant crop.

PLUMS.—The most important acquisition of the Plums, plum kind has been described and figured by Mr Hooker in the third volume of the London Horticultural Transactions, under the name of Wilmot's New Early Orleans Plum. In general habit the tree resembles the common Orleans; but the fruit ripens three weeks before that of the Orleans. Notwithstanding this early maturity of the fruit, the blossom is later of expanding than in almost any of the plum tribe. The combination of the properties of late flowering and of early ripening, must render this variety peculiarly valuable in the northern division of our island. The fruit resembles that of the Orleans, but is softer and more juicy, and of excellent flavour. The habit of the tree is vigorous and fertile.—Coe's Golden Drop is generally regarded as a new variety. The leaves of the tree are uncommonly large, and this is the most marked character of the variety. When the fruit is ripe, the pulp is of a gold yellow colour; on the side next the sun the skin is dotted with violet and crimson. The fruit may be kept for many weeks if suspended in a dry place. The tree requires a wall, but succeeds very well on a west aspect.

The Hailes Plum is a seedling of excellent qualities, which has lately been raised at Hailes, near Edinburgh, by Mr Clephane, gardener there. The foliage of the tree is remarkably light-coloured; the fruit is yellowish-white, juicy, and has a good deal of the rich flavour of the greengage.

Cherries.—To Mr Knight we are indebted for four new or seedling cherries, all of which are either good or highly promising.—1. The Elton is the production of a blossom of the graftion, to which the pollen of the white heart had been applied. This variety is distinguished by a deep tinge of crimson on the petals, and by the great length of the fruit-stalks. The pulp is very juicy, and has a delicate flavour.—2. The Black Eagle was from the graftion and the May-duke. Both tree and fruit bear a considerable resemblance to the May-duke.—3. The Waterloo (so named from the circumstance of the fruit having first ripened about the time of the celebrated battle) had the same origin. The fruit is somewhat later in ripening than the black eagle, and is rather larger and more conical at the point. It is nearly as hardy as the May-duke, and has been observed to attain tolerable perfection even in cloudy and rainy weather. When ripe, it is of a deep red colour, almost black.—4. The Early Black had also the same origin, and is nearly allied to the immediately preceding, from which it is most easily distinguished, by having a shorter fruit-stalk. It ripens fully a week before the May-duke, and is therefore one of the very earliest cherries. The pulp is soft and sweet, but not very juicy nor rich. As the original tree, however, is still very young, the fruit will in all probability improve in the qualities of juiciness and flavour.

Grapes.—For one of the best new grapes, the Variegated Chasselas, we are likewise indebted to the indefatigable Mr Knight. He procured it, by bringing the pollen of the Aleppo grape to a flower of the White Chasselas. The berries are striped and beautiful, have a thin skin, and are very juicy. The vine has been found to be very hardy, and constantly productive, bearing good crops on the open wall in England. The bunches gathered in October, and hung up in rather a damp room, may be kept till February or later.—The Pitmaston White Cluster was raised, as intimated in the name, by Mr Williams, who has already been mentioned as a very active and intelligent amateur horticulturist, in the West of England. It sprang from a seed of the Anverna, or small black cluster, the variety which is common on cottage-walls near London. The berry is round; when ripe, of an amber colour, bronzed with russet on the one side. The leaf is thin, and of a dark green colour. The vine is hardy, and a copious bearer. The berries are crowded, like those of the black cluster; but the bunches are larger, and ripen more early. It comes to perfection on the open wall in England. As it is early, and the berries are not apt to crack, it is well suited also for forcing.—The Esperione is not a new grape, but it has only of late come into particular notice. The vine is hardy, of luxuriant growth, and bears large crops: it perfects its fruit on the open wall near London, equally well as the sweet-water or white muscadine. Indeed, Mr Aiton, of the Royal Gardens at Windsor, mentions that, in unfavourable seasons, it has a decided advantage over these varieties, in being less retarded or affected by the state of the weather. It may, therefore, prove an acquisition in the northern parts of Britain. The bunches are large, and shouldered not unlike those of the black Hamburgh. The berries are of a fine dark colour, with a bluish farina; the pulp adheres to the skin; though neither highly flavoured nor melting, it is very pleasant.

Gooseberries.—Great attention has been paid, Gooseberries for a number of years past, to the raising of new and improved varieties of the Gooseberry. This being a branch of experimental horticulture fortunately within the reach of almost every man, it is pleasant to observe, that it has been practised especially by the cultivators of Lancashire, many of whom are workmen having small gardens for their recreation. For size, in particular, the gooseberries of Lancashire excel all others; insomuch that foreigners, at first sight, generally regard them as belonging to the plum tribe. To enumerate even the principal varieties seems unnecessary; numbers are constantly rising into some degree of notice, while others, of temporary celebrity, are losing ground. Among the red, the old ironmonger, the red Champagne, the Warrington, and the captain, are at present held in high esteem. Wilmot's early red likewise deserves particular mention: it is very early ripe, and of excellent flavour; in May, it is better for tarts and sauces than most others, being larger, and the skin not being tough, but melting down with the rest of the berry. The bush is easily cultivated, and is very productive.

Raspberries.—Mr Williams of Pitmaston has lately raised from the seed a double-bearing red raspberry, the fruit of which is greatly superior to that of the old double-bearing variety. The second crop of this new kind begins in the end of August, and continues till the end of October. The autumnal fruit is produced not only at the ends of the annual shoots, but also on suckers, which rise from the root about midsummer, and bear abundantly.

Currants.—Of the currant no variety superior to the large Dutch white and the Champagne has yet come into general notice. The latter is intermediate between red and white, and is larger and more juicy than the red. The Pollock white is an excellent variety, which has been raised from the seed, at the garden of Sir John Maxwell, Bart. by Mr Campbell, the gardener there, but which is not yet generally known. The property on which its excellence depends is superior sweetness. It may be remarked, that the importance of thus gaining from the seed more saccharine kinds has greatly increased, in consequence of the very general employment of the berries in the making of home made wines.

Strawberries.—Of the strawberry several new varieties have lately appeared. Of these novelties, one originally raised by Messrs Caddenhead at Aberdeen, and called the Roseberry Strawberry, has acquired the highest character for excellence. In the berry it resembles the scarlet or Virginia, but it is larger, and of a richer flavour. The flower-stalk of the plant rises completely above the leaves; the produce is very great, and the fruit ripens in succession for several weeks, in this respect resembling the habit of the alpine strawberry. So prolific is this variety, that plants which have been forced in the early spring, and yielded a crop in the hot-house, afford, when turned out of the pots into the open border, a second crop in the summer. Nay, Mr Lee of Ham- mersmith repotted in the autumn some plants which had been forced in the spring; and, on being placed in a vinery, they produced ripe fruit in November and December, not only very fine in appearance, but excellent in flavour. The roseberry is now much cultivated, both for the Edinburgh and London market.—Mr Knight has raised, from the seed of the scarlet, a variety which is now called the Downton Strawberry. The fruit is large, but irregular in shape; the external colour a bright scarlet; the pulp soft, juicy, sweet, and of a rich flavour. The plants produce abundantly, and are hardy, the leaves remaining green through the winter.—A variety, called the Mulberry Strawberry, is now likewise a good deal cultivated at Edinburgh. The berries make a fine appearance, being of a dark purplish red colour, and the acini of the fruit being large. They are not, however, desirable for the dessert, the firmness of the pulp rendering them coarse when compared with the roseberry or scarlet. For preserves they are well adapted, on account of the quality just mentioned, and also of their flavour.

AMERICAN CRANBERRY.—As a new and recent addition to our hardy cultivated fruits, the American Cranberry (Vaccinium macrocarpon) deserves particular notice. It is distinguished by the smoothness of the stems, and the largeness of its fruit. It grows freely, and produces its fruit readily, in any damp situation, though not absolutely marshy; but wherever there is a pond, it may be cultivated with the greatest success. The margin of the pond, or a part of it, if large, is to be prepared, by driving in stakes a short way within the water line; boards are so placed against these, as to prevent the soil of the cranberry-bed from falling into the water. Small stones, such as are raked from the garden borders, are laid in the bottom; and over these, peat or bog earth, mixed with sand, to the extent of about three or four inches above, and half a foot below, the usual surface of the water. If the plants be placed at six feet asunder in this prepared border, they will cover the whole superficies of it in the course of two seasons, by means of their long runners, which take root at different points. Particular attention should be paid to this circumstance, that there are two varieties of the American cranberry, one very productive of fruit, the other not so; of course, the former is to be greatly preferred. From a small space, a large quantity of cranberries may be gathered: if the bed be thirty or forty feet in length, by five or six in breadth, a quantity will be procured sufficient for the supply of a family throughout the year. The cranberries are easily preserved in bottles, till wanted for use in tarts or otherwise.

TENDER EXOTIC FRUITS.—Notice may here be taken of one or two tender exotic fruits, which have of late years been cultivated in our hot-houses.

The GRANADILLA VINE (Passiflora quadrangula- ris) is, in some places in England, particularly at Harewood House, treated as a fruit-bearing plant. The fruit, called Granadilla in the West Indies, is of a greenish-yellow colour, the size of a goose-egg, sweet, and of a very pleasant flavour. The temperature of the warmest hot-house is necessary for its production. The plant is pruned much in the manner of the grape-vine. The only peculiar part of the culture seems to consist in annually cutting-in the roots to within six inches of the stem, and giving at the same time a supply of fresh rich loam. (Lond. Hort. Trans. Vol. IV. Part I.) It is proper likewise to assist the fecundation of the germen, by drawing a camel's-hair pencil over the anthers, and applying it to the style.

The PURPLE-FRUITED PASSION-FLOWER (Passi- flora edulis) is now to be found in many stoves around fruited London, treated as a fruit-bearing plant. The pro- duce, which is ready about November and December, is abundant, and beautiful to the eye; but we cannot help thinking, that the very large space occupied by the plant might be better employed. The finest specimens of this fruit scarcely surpass in quality the common red magnum plum, to which they bear some resemblance.

The LO-QUAT (Mespilus Japonica) has for a num- ber of years been cultivated as a fruit-bearing tree in the hot-houses at the seat of Lord Bagot in Staffordshire. The mode of culture adopted by his Lord- ship is described in the third volume of the London Horticultural Transactions. The plants, which are kept in large pots, and are six or seven feet high, are set out of doors from the middle of July till the middle of October, thus imitating the winter of their native climate. They are then removed into the warmest situation in the stove. They flower in De- cember, and ripen their fruit about March. The fruit is much esteemed in the East Indies; but a gentleman who had eaten it in Ceylon, gave the pre- ference to that produced in our hot-houses. The cultivation of the lo-quat is extending, the plant already existing in many collections where it has ne- ver been treated as a fruit-bearing tree.

The Orangery.

It may be doubted if more attention be now paid to the orangery than in former times. Per- haps the number of large orange trees in the coun- try has rather declined. Still, however, their cul- ture has in some places been improved, and kinds are now cultivated with success which were formerly little known. The Citron and the Lemon are more hardy than the Orange; and the former are now therefore preferred for training on trellises, or for covering the back wall of a hot-house. Mr Benham at Isleworth near London, and Mr Hen- derson at Woodhall near Glasgow, are, we believe, among the most successful cultivators of the orange tribe, in this country. The Malta orange, or Sweet Philippine orange, has lately been introduced. It Flower Garden and Shrubbery.

Under this head, the improvements may be considered as consisting chiefly in the introduction of various ornamental shrubs and flowers, formerly unknown to our pleasure-grounds and parterres. All that here seems necessary, therefore, is to mention the most important of these, and to take notice of any peculiarities in their culture.

The first place is perhaps due to several new species of Rose from China, which have of late years added wonderfully to the beauty and richness of our flower gardens. 1. The Blush China-Rose (Rosa Indica) is so hardy, that it often unfolds its elegant pale red flowers early in the spring, notwithstanding the ungenial weather which we generally experience at that season of the year, and it continues displaying a succession of flowers till November. It is almost without scent; but the flowers are very showy and produced in great profusion. There is a sweet-scented variety, which is of dwarfish stature and not so hardy. When this is placed in a conservatory, it proves highly grateful by its odour, as well as ornamental by its delicate colour: it should, however, be observed, that there are two sorts of this; one having a much richer perfume than the other. 2. The Crimson China-Rose (R. sempervirens) is an elegant spreading shrub. It requires a sheltered situation, and in general succeeds best when trained on the outside of a green-house or hot-house wall. Some varieties with semi-double flowers are extremely beautiful, and worthy of a place in the conservatory. 3. The Macartney Rose (R. bracteata), although neither so hardy nor so beautiful as the preceding, tends also to decorate the exterior of our hot-houses with its milk-white flowers, during the greater part of the summer. 4. The Bramble-flowered Rose (R. multiflora) requires to be trained against a wall with a southern aspect: here, however, it often proves very ornamental, the flowers coming forth in large clusters. 5. Lady Banks's Rose (R. Banksiae) is remarkable for the elegance of its foliage; and it is hardy, growing pretty freely in our open borders, and producing its blossoms readily.

A white variety of the Moss Rose has of late attracted much notice, on account of its variety and uncommon appearance. It does not appear to be very permanent, but rather apt to return to the usual hue.

The Ayrshire Rose (R. capreolata of Don) has likewise excited a good deal of attention. It grows with great rapidity, and has been found very useful for covering any offensive wall, paling, or roof. There are two kinds; the one most commonly sold in the public nurseries is merely Rosa arvensis, a native of this country: the other is more nearly allied to R. sempervirens, a native of the south of Europe; from which, however, it differs considerably in habit; in particular, the Ayrshire rose is more hardy, and grows more freely, and during winter it does not retain its leaves nearly so much as the sempervirens.

Many varieties of the Scots Rose (R. spinosissima) have been raised; some double, others semi-double, but variously coloured in the petals. These, it may be remarked, naturally flower early in the summer; and it seems reasonable, therefore, to regard them as well adapted for forcing in the spring.

Some very ornamental Japan shrubs are particularly deserving of notice. 1. The Corchorus Japonicus (or Kerrea Japonica), trained against a north or east wall, retains its leaves through the winter, and early in the spring produces its rich yellow blossoms in profusion. 2. The Japan Apple (Pyrus Japonica), trained on the outside of a green-house or hot-house, displays at the same early season its beautiful red blossoms. A white-flowered variety has likewise been introduced, and forms a good contrast with the other. In favourable situations the fruit often attains a considerable size during summer; but the shrub is of importance only in the way of ornament. 3. The Gold Plant of Japan (Aucuba Japonica) highly adorns the shrubbery, especially during winter, by its brilliant leaves, blotched with gold yellow. In sheltered situations it sustains our ordinary winters without injury.

A variety of the lilac-tree, apparently a hybridous Siberian production between the common and the Persian, Lilac. It is now cultivated, under the name of Siberian Lilac. It forms a pretty shrub, the size of the leaf being intermediate between that of the two old species. It seems to be the Varin of the French.

Ribes aureum, or the Yellow-flowered Currant, Yellow Currants makes a fine appearance when covered with its blossoms in May. It requires a sheltered place, or to be trained to a wall.

Ireland has, within these few years, produced three very ornamental varieties, all of them evergreens. 1. The most important is a broad-leaved ivy, usually distinguished by the name of Irish Ivy. For all the purposes for which ivy is desirable in a garden, this kind is preferable. It not only grows more freely, but its leaves are four times larger than in the common ivy, and of a brighter green. For 'decorating or disguising the back of a wall, it is well calculated, on account of the beauty of the leaves; for covering a rock or an aged tree in the pleasure grounds, it is equally adapted. 2. The next is a kind of Yew-tree, first observed at Florence Court, remarkable for its upright growth, and commonly distinguished by the name of Irish Yew. It forms a very fine object in the shrubbery, its dark foliage contrasting with the light hue of the cypress or the Swedish juniper. It is so different in aspect from the common yew, that some regard it as a distinct species. 3. The Irish Furze is a very recent production, and it is likewise remarkable for its upright growth. For small cross hedges, or brise-vents, in flower-borders, it is very desirable, being at the same time curious and ornamental, and completely answering the purpose. It is propagated by cuttings, but these do not strike very readily.

For the many fine flowering shrubs which require a bog soil, compartments are now prepared with great care, generally in a low situation, or by the side of a rivulet or pond. Surface peat-earth, having a considerable portion of fine sand intermixed with it, forms the most desirable soil. From the circumstance of several of the most showy plants (particularly the whole genus Kalinia, with different species of Azalea and Andromeda, and one fine Rhododendrum, R. maximum), being natives of America, these compartments are generally called the American grounds. The list of plants adapted to those grounds has been considerably increased. Rhododendrum Catawbiense, Caucasicum, and Dauricum, may be particularly specified, with Andromeda pulverulenta and cassinefolia; and some new varieties of Azalea nudiflora and viscosa.

The Tree-peony or Moutan, if planted in a sheltered situation in the garden, and protected by a temporary cover during winter, forms a most beautiful ornament when in flower in the beginning of summer.—This may be considered as a connecting link, leading from the notice of shrubs to herbaceous plants. But here brevity must be studied; and only a few of the most ornamental can be named.

The different species and varieties of herbaceous Peonies may first be noticed, these having, of late, been much in vogue, and getting in many places a separate border of the flower-garden allotted to them. The following are at present cultivated: P. corallina; paradoxa fimbriata, or double-fringed; peregrina compacta, or byzantina ; albiflora, in three subvarieties, with single flowers, with double flowers, and double sweet-scented (the latter one of the finest); daurica ; tenuifolia, anomala, or genuine laciniiata ; albiflora Tatarica, or Sibirica ; officinalis with double red flowers (one of the oldest inhabitants of our gardens): the same with double flesh-coloured flowers, and a still paler variety approaching to white.

The cultivation of Dahlias has become fashionable, and they must not, therefore, be omitted.—There are two species, D. superflua and D. fruticosa. Of the former there are purple, scarlet, and rose-coloured varieties; of the latter, saffron-coloured and white. Occasionally most of these are procured with double or semi-double flowers, and these are most highly prized by florists. The roots, which are tuberous like those of the common peony, are taken up in autumn, and kept in a dry place, beyond the reach of frost, till the time of planting in the spring. Burying them among sand is unnecessary, and often proves hurtful. In April the more choice kinds should be planted in pots, so as to have their growth forwarded in a frame or green-house. In June they may be planted out : a rich border is not desirable for them; on the contrary, the flowers come more brilliant in a poor soil. If the plants show a great disposition to be luxuriant, the flowering is impaired: this disposition may be somewhat checked, by pinching off some of the secondary branches while young and tender. After the flower-bud has appeared little water should be given to the plants, even though the weather should prove dry.

The Cardinal flower (Lobelia cardinalis) was long Lobelias admired; but it is surpassed by two species lately introduced, Lobelia fulgens and L. splendens. These are fortunately more hardy, or at least more easily kept than the former. In mild winters they stand perfectly well in the open borders; but the stools should be separated in the spring, the young slips forming much finer plants. It may be proper, however, to preserve two or three well-established plants of each kind in pots in the green-house during winter, and to divide the sets in spring at the time of planting out. A compartment in the flower-garden filled with these, makes a most brilliant appearance in the months of August and September. If it is wished to see them in full luxuriance and splendour, more care is requisite. The offsets should be potted in October, and kept in a frame or cool greenhouse till the spring, when they should be repotted and subjected to increased temperature. During their growth they should be kept very moist, perhaps even with a pan of water under the pot. Treated in this way, they become as strong and tall as plants of the Pyramidal Bell-flower (Campanula pyramidalis); and as they produce their flowers at the same period, the blue and crimson form a fine contrast.

The Tiger-spotted Lily of China (Lilium tigrinum) Tiger Lily. is a valuable acquisition, being quite hardy, and, when planted in a considerable clump, becoming extremely ornamental. It succeeds well in soil prepared with a portion of bog-earth, somewhat in the manner of the American ground. The bulbs may be left in the ground without risk of injury, unless the situation be very damp. They multiply rapidly at the root, however, by means of offsets; and the roots must, therefore, be occasionally parted. The plant is also readily propagated by means of the small bulbs produced in the axillae of the leaves.

The Mexican Tiger-flower (Tigridia pavonia) succ. Tiger-flower. seeds pretty well in the front of a hot-house, and for several weeks expands daily some of its most gorgeous, but transitory flowers. The roots require to be lifted at the approach of winter, and to be kept carefully from the access of frost.

Great attention has, for some years past, been paid to the important subject of rendering the plants of warmer countries sufficiently hardy to enable them to sustain our variable climate. The most effectual way is to endeavour to bring such plants to ripen their seeds in the open air in this country with as little assistance from glass as possible; and then to sow these seeds, from which a somewhat more hardy progeny may be looked for. By continuing this mode for several successive generations, the plant may (according to the theory of Sir Joseph Banks, Flower Gar.-when treating of the Canada rice) be completely den and naturalized.

Green-house and Conservatory. The additions to the ornamental inhabitants of the green-house or the conservatory have, of late years, been very great. We can only notice a few of them, and these very generally.

Cape Heaths. The Heaths of the Cape of Good Hope have proved so numerous, and, at the same time, so beautiful, that, in many places, a separate green-house has been established for them, under the name of the Heathery. About 240 species are now cultivated; and they are highly worthy of the care and expense bestowed on them, some species or other being in flower in almost every month of the year, and several of them being fragrant. Most of them have been figured by Mr Andrews, in a splendid work, entitled, Engravings of Heaths, with Botanical Descriptions. All the ericae grow best in a mixture of bog-earth and sand. They require as much free air during winter as can be given to them, without absolutely subjecting them to frost. They are generally propagated by cuttings, as many species do not produce their seeds in this country; and he is accounted an expert propagator, who succeeds readily in striking cuttings of Erica ardens, taxifolia, Massoni, retorta, articulata, and elegans.

Geranium. The number of showy Geraniums has greatly increased; the raising of seedling varieties having for some years been a favourite occupation of florists. Some of the finest are varieties of Pelargonium-inquinans, with flowers of an intensely crimson colour, and with semi-double flowers; others with large blossoms, finely marked on a light ground, have sprung from P. cucullatum.

Tree Mignonette. A simple but desirable addition to the ornament of the green-house or the lobby must not be passed over. A variety of Mignonette has been introduced, which, when kept in pots, remains in flower, and, what is more important, in full fragrance, throughout the winter. By training, it is made to assume somewhat the shrubby appearance, and is called Tree Mignonette. It seems to be a variety very distinct from the common kind; the leaves are much smaller, and the flowers are produced in greater abundance.

The Dry Stove has received rich accessions to its treasures, in numerous new species of Stapelia and Mesembryanthemum.

Acacias. The Conservatory is now filled with the curious and beautiful Acacias of New Holland. These are not less remarkable for their singular foliage (which is generally upright, and acts equally on light in every direction, or, in other words, the leaves have no upper and under surface), than for the profusion in which they display their rich yellow flowers in the spring season.

The Japan Rose (Camellia Japonica) must not be forgotten; for some most beautiful varieties, forming the pride of the conservatory, have appeared within the last few years; particularly the waratah or anemone-flowered, both with double red flowers and with double white flowers, the latter often called the Pompon camellia.

The Hydrangea hortensis succeeds in the open border in good seasons; but it is always much injured during winter. If room can be spared in the conservatory, it makes a much finer appearance there; and, according to Mr Hedges (Lond. Hort. Trans. Vol. III.), by planting it in pure yellow loam, the flowers may be procured of the beautiful blue colour sometimes observed in this species,—an experiment, however, which does not always succeed.

Tender aquatics, of the genera Nymphaea, Nuphar, Aquaticas, Menyanthes, and Nelumbium, are now cultivated with great success in frames resembling those used for the raising of melons. The plants are placed in cisterns, made of wood and lined with lead, about four feet in length, and two feet and a half in breadth; and these cisterns are plunged in tanners' bark or stable dung. In this way the plants flower much more freely and beautifully than when kept in lofty hot-houses heated by flues.

From the details which have been given, it pretty clearly appears, that great advances are making in the knowledge and practice of an improved horticulture. For this fortunate state of matters, we are in no small degree indebted to the two patriotic associations, the Horticultural Society of London and the Horticultural Society of Edinburgh, already mentioned. Both consist of several hundred members, all of them amateurs of gardening. Among them, horticultural knowledge must rapidly increase, and a beneficial feeling of friendly emulation cannot fail to be excited. Both societies distribute honorary rewards for excellence in any of the productions of the garden, or for the encouragement of well-contrived experiments. Both publish Transactions, which have been repeatedly quoted in the preceding pages; thus affording equally to the scientific cultivator and to the practical gardener a convenient medium for communicating to the public notices of useful improvements. The London Society has already given to the world three volumes in quarto, embracing many important subjects, some of them illustrated by engravings in the first style of excellence. The Scottish Society has published two volumes in octavo, likewise meritorious as to matter, but with slender pretensions to ornament or illustration.

(G.G.G.) ADDENDUM

TO

VOLUME FOURTH.*

EQUATIONS.

1. In all the applications of Algebra, it is not the magnitudes concerned that we immediately consider, but merely their proportions. In every class of quantities of the same kind, one being adopted as the unit of comparison, all the rest are referred to this standard, and are represented by the proportions they bear to it. The letters of the alphabet, or other symbols used in Algebra, are not, therefore, properly speaking, the representatives of magnitudes; they denote ratios, or abstract numbers, viewed, as in the fifth book of Euclid, in the most general manner, and independently of any particular system of arithmetic or numeration.

The ancient Geometry follows a different procedure. In that science the attention is in every case confined to the magnitudes under actual consideration. A general property of triangles is established, by showing, that it is true of any particular triangle that comes under the proposed hypothesis. The geometer contemplates particular instances, presenting, for the most part, relations not very complex, and easily kept in view. On this account he carries on his investigations with the greatest clearness, and is in no danger of falling into contradiction or paradox. But his science is little susceptible of general methods. If any process within the compass of the ancient geometry be entitled to that appellation, it is what is called the method of exhaustion. Every geometer perceives that all the demonstrations under this head have the closest analogy. Yet, after a hundred applications, it is still necessary, in any new case, to pursue the reasoning through all its details, without deriving assistance from any general conclusion previously obtained.

Algebra possesses a great advantage over geometry in generalizing its processes. Problems relating to magnitudes of the most different kinds, nevertheless, lead to similar expressions in numbers. Questions in geometry, in mechanics, or concerning mercantile business, are made to depend on the same rules for their solution. It may be said that algebra and the modern analysis accomplish, for all the mathematical sciences, the project, entertained by some ingenious men, of an universal and philosophical language, which, being founded on an exact scrutiny into the nature of things, and on what they possess in common, might greatly facilitate the acquisition and the extension of our knowledge.

The spirit of generalization peculiar to algebra is no where more conspicuous than in the doctrine of equations. Every determinate problem that can occupy the attention of the mathematician, is ultimately reduced to the finding of such numbers as are necessary to determine the unknown quantity or quantities, by means of the equations that subsist between those numbers, and others which are given in the question. A wide field of mathematical investigation is thus brought under a limited number of algebraic expressions.

In treating of equations it will not be necessary to begin with laying down a formal definition. We confine ourselves, in this article, to the consideration of such equations as contain only one unknown quantity. We further suppose, that the elementary operations preparatory to solution are already performed; so that the unknown quantity is clear of radical signs, and is no where found in the denominator of a fraction: likewise that all the separate terms are brought to one side of the sign of equality, and arranged in such a manner, that the first term, which must always be positive and have unit for its index, contains the highest power of the unknown quantity or \( x \); the second term contains the next highest power, and so on, the term which does not contain \( x \) being placed last. This arrangement must always be understood when any term is distinguished by the order it stands in; but it will sometimes be convenient to write the terms in an inverted order, arranging them according to the indices of the unknown quantity.

Equations are divided into different classes or orders, according to the highest power of the unknown quantity found in their terms.

An equation of the first degree, or a simple equation, is one which contains \( x \) only, without any of its powers, as \( x - A = 0 \).

A quadratic equation, or one of the second degree, contains the square of \( x \), as \( x^2 - A = 0 \), or \( x^2 - Ax + B = 0 \).

A cubic equation, or one of the third degree, contains the cube, or third power of \( x \), as \( x^3 - A = 0 \), or \( x^3 - Ax^2 + Bx - C = 0 \).

A biquadratic equation, or one of the fourth degree, contains the fourth power, or biquadrate, of \( x \), as \( x^4 - A = 0 \), or \( x^4 - Ax^3 + Bx^2 - Cx + D = 0 \).

And, in general, an equation of the \( n \)th degree contains the \( n \)th power of \( x \), and the powers inferior to the \( n \)th, such as

\[ x^n - Aa^{n-1} + Bx^{n-2} \ldots \ldots \ldots - Mx + N = 0. \]

A root of an equation is a value of the unknown number \( x \). Thus, if \( a \) represent a number, and if its powers, \( a, a^2, a^3, \) &c. when they are substituted in the equation for \( x, x^2, x^3, \) &c. produce an equality between the positive and negative terms, then \( a \) is a root of the equation, and it is a positive root; but if, for \( x, x^2, x^3, \) &c., we must substitute \( -a, a^2, -a^3, \) &c., which are the powers of \( -a \), in order to obtain the like equality, then \( a \) is a negative root of the equation.

What we have here called roots are more generally named real roots, to distinguish them from those expressions to which the appellation of imaginary or impossible roots has been given. As it will conduce to perspicuity, we shall always use the word root in the sense here defined, unless when imaginary or impossible roots are expressly mentioned.

From the definitions laid down, it follows that the negative roots of the equation,

\[ 0 = N + Mx + Lx^2 + Kx^3 + \text{&c.} \]

are the same with the positive roots of the equation,

\[ 0 = N - Mx + Lx^2 - Kx^3 + \text{&c.} \]

in which the signs only of all the terms containing the odd powers of \( x \) are changed. For the same result is obtained, whether we make \( x \) equal to \( -a \) in the first equation, or to \( +a \) in the second.

2. A great advantage has resulted from the practice introduced by Harriot, of writing all the terms of an equation on one side of the sign of equality. The polynomials formed by all the terms thus brought together are rational and integral functions of the unknown quantity; and the question is, to find in what circumstances such expressions are equal to zero. The most likely way of succeeding in this research, is to resolve the functions into their most simple component factors. Harriot supposed that every rational function can be produced by the continued multiplication of binomial factors; and, in this, he has been followed by succeeding algebraists. The modern theory of equations is entirely founded on this supposition, which, although it has not been demonstrated, has yet, in some measure, been verified in the progress of the science, and by the admission of those artificial expressions called imaginary or impossible quantities. But there is a distinction between the real and impossible binomial factors of a rational polynomial. For the first are expressions complete and significant by themselves, without reference to other quantities; whereas one impossible factor necessarily supposes the existence of another, the two related expressions being such, that their multiplication produces one real factor of the second degree. Thus, every pair of impossible factors is equivalent to a real quadratic factor; and, by an unavoidable consequence of the forced supposition made by Harriot, the attention of algebraists has been drawn to the two impossible expressions, instead of being directed to the real one which they compose. In order to place the doctrine of equations and the theory of impossible roots on a solid foundation, it appears necessary to attempt the resolution of rational functions into their component factors by a rigorous analysis, free from arbitrary suppositions.

To resolve the rational function \( f(x) \) into its component factors, we must begin with inquiring, whether it can be divided without a remainder, by a division such as \( x - a \), or \( x + a \)? If it can, the proposed function will be equal to \( (x-a) \times f'(x) \), where \( f'(x) \), the quotient of the division, is a function similar to \( f(x) \), but of an order one degree lower. In like manner, it may be possible to reduce \( f'(x) \) to a degree still lower, by means of one or more divisors of the same form; and, in certain cases, the first function may be entirely exhausted by successive binomial divisors. When this happens, the divisors \( x - a, x - b, x - c, \) &c. will be equal in number to the exponent of the highest power of \( x \), and their continued product will be equal to \( f(x) \). It is evident, that by multiplying together a proper number of such factors, an algebraic expression may be formed similar to any rational and integral function, and the coefficients of this product will likewise contain as many quantities to be determined at pleasure as there are coefficients in the given function. But we should reason badly if, from this process of composition, we should infer that a product arising from the multiplication of a certain number of simple factors may have any given coefficients, or will coincide with any proposed polynomial of the same degree. This is a point that can be ascertained only by a process of analysis or resolution, and by seeking all the binomial divisors any given function admits of. In fact, the cases are extremely rare in which an algebraic function can be completely exhausted by real binomial divisors. There are many polynomials which have not a single divisor of this kind; and, in the progress of resolution, we generally arrive at a function which cannot be further divided. When this is the case, it must be tried whether a quadratic divisor, as \( x^2 + mx + n \), will not be successful in lowering the function. But here it must be observed, that such divisors are of two kinds; one, as \( (x - \xi)^2 - r^2 \), which can be resolved into two binomial factors; and one as \( (x - \xi)^2 + r^2 \), which cannot be so resolved without introducing imaginary or impossible expressions. Now, to divide by a divisor of the first kind is the same thing as to divide by the two binomial factors of which it is composed; and, therefore, it is the second kind of quadratic factors only that need be tried, or that can succeed, in lowering a function already deprived of all its simple divisors. After quadratic divisors those of the third degree would naturally come to be considered; but this is unnecessary, because algebraists have found that every rational function may be completely exhausted by simple and quadratic factors.

What has now been said naturally distributes the subject under two heads; one treating of the simple or binomial factors, and the other of the quadratic or trinomial factors, of algebraic equations.

Binomial Factors.

3. The first object of inquiry must be to find the conditions necessary, in order that a binomial quantity, as \( x - a \), or \( x + a \), shall divide a rational polynome without a remainder. Suppose that \( x - a \) is a divisor of the polynome,

\[ x^n + Ax^{n-1} + Bx^{n-2} \ldots + Mx + N, \]

which we shall denote by \( f(x) \): then we shall have

\[ f(x) = N + Mx + Lx^2 + Kx^3 + \&c. \] \[ f(a) = N + Ma + La^2 + Ka^3 + \&c. \]

wherefore, by subtracting and dividing by \( x - a \), we get

\[ \frac{f(x)}{x-a} - \frac{f(a)}{x-a} = M \frac{x-a}{x-a} + L \frac{x^2-a^2}{x-a} + K \frac{x^3-a^3}{x-a} + \&c. \]

Now, it is known, that the difference between any like powers of two numbers is exactly divisible by the difference of those numbers; hence all the quantities on the right-hand side of the sign of equality form an integral expression. But as \( f'(a) \) does not contain \( x \), it cannot be divisible by \( x - a \); it follows, therefore, that \( f(x) \) cannot be divisible by \( x - a \), unless \( f(a) = 0 \); and it is obvious, that this condition is the only one necessary. Thus, the polynome \( f(x) \) will be divisible by \( x - a \), when \( a \) is a positive root of the equation \( f(x) = 0 \); otherwise not.

Again, let the divisor be \( x + a \): then,

\[ f(x) = N + Mx + Lx^2 + Kx^3 + \&c. \] \[ f(-a) = N - Ma + La^2 - Ka^3 + \&c. \]

and, by proceeding as before,

\[ \frac{f(x)}{x+a} - \frac{f(-a)}{x+a} = M \frac{x+a}{x+a} + L \frac{x^2-a^2}{x+a} + K \frac{x^3+a^3}{x+a} + \&c.; \]

here again all the divisions on the right-hand side of the sign of equality can be exactly performed: and we must, therefore, conclude that \( f(x) \) will be divisible by \( x + a \) only when \( f(-a) = 0 \), that is, when \( a \) is a negative root of the equation \( f(x) = 0 \).

Now \( x - a \) being a divisor of \( f(x) \), the quotient, which we may denote by \( f'(x) \), will be a polynome of \( (n-1) \) dimensions, or one degree lower than \( f(x) \): and we shall have

\[ f(x) = (x - a) \times f'(x). \]

From this equation, it appears that every value of \( x \) that makes \( f'(x) \) equal to zero, will likewise make \( f(x) \) equal to zero: consequently, every binomial divisor of the first function will likewise be a divisor of the second. And, if \( f'(x) \) has no roots, and no binomial divisors, neither will \( f(x) \) have any roots except \( \pm a \), nor any binomial divisors except \( x \mp a \). Suppose that the polynomes \( f(x) \) and \( f'(x) \) have the common root \( \pm b \); they will likewise have the common divisor \( x \mp b \); and if we put \( f''(x) \) for the quotient arising from the division of \( f'(x) \) by \( x \mp b \), so that \( f'(x) = (x \mp b) \cdot f''(x) \); we shall have

\[ f(x) = (x \mp a) \cdot (x \mp b) \cdot f''(x), \]

in which equation \( f''(x) \) is a polynome of \( n-2 \) dimensions, or two degrees lower than \( f(x) \).

It is evident, we may continue to reason in the same manner, either till, after successive divisions, we come at last to a binomial quotient, in which case the original polynome \( f(x) \) will be completely resolved into binomial factors; or till we come to a quotient that has no roots, in which case \( f(x) \) will have no binomial factors except those previously found. We may, therefore, conclude that "a rational polynome has as many binomial factors as it has roots, and no more; every positive root producing a factor of the form \( x - a \), and every negative root one of the form \( x + a \); and since the number of binomial factors can never be greater than the dimensions of the polynome, its roots cannot exceed the same number."

4. There are very few cases in which it can be known immediately and by inspection, that an equation has one or more roots. These cases depend upon the following proposition, viz. "If \( \varphi(x) \) denote a rational polynome having \( x \), or some integral power of \( x \), in every one of its terms, and likewise having the term that contains the greatest power of \( x \) positive, a value of \( x \) may be found that will make \( \varphi(x) \) equal to any positive quantity, as \( s \)."

Suppose, first, that all the terms of \( \varphi(x) \) are positive; then, \( x^n \) being the first term, or that in which \( x \) rises to the highest power, if \( s = t^n \), and \( \lambda > t \), it is manifest, that

\[ \varphi(\lambda) > t^n > s. \]

Therefore, while \( x \) increases from 0 to be equal to \( \lambda \), the function \( \varphi(x) \) increases from 0 to be greater than \( s \); and as the variations of \( \varphi(x) \), however irregular, Equations. they may be, are connected by the law of continuity, the function will pass through every gradation of magnitude between o and the greatest limit \( \varphi(\lambda) \). Consequently, there is a value of \( x \) between o and \( \lambda \), that will make \( \varphi(x) \) equal to s.

When the terms of \( \varphi(x) \) are not all positive, let all the positive terms except \( x^n \) be rejected, and all the negative terms be retained, and we shall have \( \varphi x \) equal to, or greater than,

\[ x^n - Fx^{n-1} - Hx^{n'-1} - \&c. \]

But, s being equal to \( t^n \), we have

\[ t^n = x^n - (x-t) \cdot \left\{ x^{n-1} + tx^{n-2} + t^2x^{n-3} \ldots + t^{n-1} \right\}. \]

Now, by equating the negative terms of the first expression to the terms containing the like powers of \( x \) in the value of \( t^n \), we shall get

\[ (x-t) \cdot t^i = F \] \[ (x-t) \cdot t^j = H \] \& c.

And hence,

\[ x = t + \frac{F}{t^i} \] \[ x = t + \frac{H}{t^j} \] \& c.

Let \( \lambda \) be either equal to, or exceed the greatest of these values of \( x \); then we shall have

\[ \varphi(\lambda) > t^n > s. \]

Wherefore, as before, there is a value of \( x \) between o and \( \lambda \), that will make \( \varphi(x) \) equal to s.

From what has now been proved, we derive the following properties of equations.

1. "Every equation of odd dimensions has at least one positive root when the last term is negative, and one negative root when the last term is positive."

If the last term be negative, as in this instance,

\[ x^{2n+1} + Ax^{2n} + Bx^{2n-1} \ldots + Mx - N = 0; \]

according to what has been proved, a value of \( x \), viz. \( a \), may be found that will satisfy the condition,

\[ a^{2n+1} + Aa^{2n} + Ba^{2n-1} \ldots + Ma = N; \]

then \( a \) is a positive root of the equation.

When the last term is positive, as in this equation,

\[ x^{2n+1} + Ax^{2n} + Bx^{2n-1} \ldots + Mx + N = 0; \]

change the sign of the last term, and the signs of all the terms that contain the even powers of \( x \), then the polynome will become

\[ x^{2n+1} - Ax^{2n} + Bx^{2n-1} \ldots + Mx - N: \]

and a value of \( x \), viz. \( a \), may be found such that

\[ x^{2n+1} - Aa^{2n} + Ba^{2n-1} \ldots + Ma = N: \]

now transpose N, and then change the signs of all the terms, and we shall get

\[ -a^{2n+1} + Aa^{2n} - Ba^{2n-1} \ldots - Ma + N = 0, \]

which shows that \( a \) is a negative root of the equation.

2. "Every equation of even dimensions having its last term negative, has two roots, one positive and one negative."

Let the equation be

\[ x^{2n} + Ax^{2n-1} + Bx^{2n-2} \ldots + Mx - N = 0; \]

and consider the polynomials,

\[ x^{2n} + Ax^{2n-1} + Bx^{2n-2} \ldots + Mx - N, \] \[ x^{2n} - Ax^{2n-1} + Bx^{2n-2} \ldots - Mx - N, \]

in the latter of which the signs of all the terms containing the odd powers of \( x \) are changed; then there are two values of \( x \), viz. \( a \) and \( b \), such as to answer the conditions,

\[ a^{2n} + Aa^{2n-1} \ldots + Ma = N \] \[ b^{2n} - Ab^{2n-1} \ldots - Mb = N: \]

consequently \( a \) is a positive, and \( b \) a negative root of the equation.

3. "A polynome of even dimensions, which has no binomial factors, is always positive, whatever value be substituted for the unknown quantity."

Let the polynome be \( f(x) \), or

\[ x^{2n} + Ax^{2n-1} \ldots Mx + N: \]

then the last term, or that term which does not contain \( x \), must be positive; for, otherwise, the polynome would have two roots, and two binomial factors, contrary to the hypothesis. Now, if it be possible, let the polynome have a negative value when \( \lambda \) is substituted for \( x \), so that \( f(\lambda) = -P \); therefore, when \( x = o, f(x) \) is equal to the positive quantity \( N \); and, when \( x = \lambda \), the same function is equal to \( -P \); but since \( f(x) \) passes through all degrees of magnitude between \( N \) and \( -P \), while \( x \) varies from o to \( \lambda \), it will become equal to zero when \( x \) has some intermediate value; therefore the polynome has one root between o and \( \lambda \), and one binomial divisor corresponding to that root contrary to the hypothesis.

It may be observed, that the converse of this proposition is not true; for a polynome of even dimensions, that has such factors as \( (x-a)^2, (x-a)^4, (x-a)^{2m} \), may never become negative, although it is capable of being equal to zero.

5. The properties demonstrated in the last section lead to this general proposition relating to the number of roots in any equation, viz. "In any equation, the number of all the roots is even when the dimensions are even, and odd when the dimensions are odd."

For every equation has as many binomial divisors as it has roots; and if we suppose an odd number of roots in an equation of even dimensions, or an even number in one of odd dimensions, the last quotient, after dividing successively by all the divisors, would be a polynome of odd dimensions, having at least one root, which would likewise be a root of the proposed equation. Therefore the number of all the roots of an equation cannot be even when the dimensions are odd, nor odd when the dimensions are even. And again, since every polynome is equal to the continued product of all its binomial divisors, and the quotient last found, after dividing by them all successively, we obtain the following proposition, viz.: "Every rational polynome is equal either to the continued product of as many binomial factors as it has dimensions; or to the continued product of an even or odd number of such factors, according as the dimensions of the polynome are even or odd, and a polynome of even dimensions, which, having no binomial factors, is always positive, whatever value be substituted for the unknown quantity."

6. When several of the binomial factors of an equation are equal to one another, it is said to have so many equal roots. In this case, the equation can be divided a number of times successively by the same binomial divisor. Thus, an equation which is twice divisible by \( x-a \), or, which is the same thing, once by \( (x-a)^2 \), has two roots equal to \( a \); and, if it can be divided by \( (x-a)^m \), it has \( m \) roots equal to \( a \).

The most obvious way of finding the conditions on which the equality of the roots depend would, therefore, be to expand the divisor \( (x-a)^m \) by the binomial theorem, and then divide the equation by it: for, after the integral quotient is obtained, the required conditions will be found by making the several parts of the remainder separately equal to zero. The number of the conditions found in this manner is equal to the exponent of the divisor; for of so many parts will the remainder of the division consist. But, in a complex operation, it is difficult to ascertain the remainder; and besides, it is not necessary to consider all the equations obtained by this process, because both the number and the value of the equal roots can be found by means of two of them only.

The inconveniences, just mentioned will be avoided by proceeding in the following manner: Let the equation be

\[ x^n + A x^{n-1} + B x^{n-2} \ldots + M x + N = 0: \]

then, if it be divisible by \( (x-a)^m \), the quotient will be a polynome of \( n-m \) dimensions; and we may, therefore, suppose that the expression

\[ x^n + A x^{n-1} + B x^{n-2} \ldots + M x + N, \]

is equal to the product,

\[ (x-a)^m \times \left\{ x^{n-m} + A' x^{n-m-1} + B' x^{n-m-2} + \text{&c.} \right\}. \]

In these expressions, \( x \) may have any value whatever; and, therefore, the equality between them will still subsist if we substitute \( x+i \) for \( x \), \( i \) being any arbitrary number; therefore the expression

\[ (x+i)^n + A(x+i)^{n-1} + B(x+i)^{n-2} \ldots + M(x+i) + N, \]

will be equal to the product,

\[ (x-a+i)^m \times \left\{ (x+i)^{n-m} + A'(x+i)^{n-m-1} + \text{&c.} \right\}. \]

Now, let the several powers of \( (x+i) \) be expanded by the binomial theorem, and put

\[ X = x^n + A x^{n-1} + B x^{n-2} \ldots + M x + N, \]

\[ Y = n x^{n-1} + (n-1) A x^{n-2} + (n-2) B x^{n-3} \ldots + M, \]

\[ Z = n \frac{n-1}{2} x^{n-2} + (n-1) \frac{n-2}{2} A x^{n-3} + \text{&c.}, \]

\[ V = n \frac{n-1}{2} \frac{n-2}{3} x^{n-3} + (n-1) \frac{n-2}{2} \frac{n-3}{3} A x^{n-4} + \text{&c.} \quad \text{&c.} \qquad \text{&c.} \]

then the given polynome of \( n \) dimensions will become

\[ X + Y . i + Z . i^2 + V . i^3 + \text{&c.} \quad (\text{A}). \]

And if the like operations are performed in the polynome of \( n-m \) dimensions; and \( (x-a+i)^m \) be expanded by the binomial theorem; the product of these two expressions will become

\[ \left\{ (x-a)^m + m (x-a)^{m-1} i + m \frac{m-1}{2} (x-a)^{m-2} i^2 + \text{&c.} \right\} \times \left\{ X' + Y' . i + Z' . i^2 + \text{&c.} \right\}. \quad (\text{B}) \]

The expression (A) being equal to the product (B), whatever \( i \) stands for, the coefficients of the like powers of \( i \) must be equal; and hence, by equating the terms in which \( i \) is wanting, and likewise the terms that contain the first power of \( i \), we get

\[ X = (x-a)^m X' \] \[ Y = (x-a)^m Y' + m (x-a)^{m-1} X'; \]

which proves that \( (x-a)^{m-1} \) is a common divisor of \( X \) and \( Y \). If, therefore, by means of the usual process, we seek the greatest common measure of the two polynomes, \( X, Y \), or,

\[ x^n + A x^{n-1} + B x^{n-2} \ldots + M x + N, \] \[ n x^{n-1} + (n-1) A x^{n-2} + (n-2) B x^{n-2} \ldots + M; \]

we shall obtain the factor \( (x-a)^{m-1} \); and the given polynome \( X \) will be divisible by \( (x-a)^m \); that is, it will contain the common factor \( x-a \) once more than the polynome \( Y \) contains it.

If we proceed farther, and equate the coefficients of \( i^2 \) in the expressions (A) and (B), we shall get

\[ Z = (x-a)^m Z' + m (x-a)^{m-1} Y' + m \frac{m-1}{2} (x-a)^{m-2} X'; \]

which shows, that \( Z \) is divisible by \( (x-a)^{m-2} \). In the same manner, it may be proved, that \( V \) is divisible by \( (x-a)^{m-3} \), and so on. It appears, therefore, that the first \( m \) coefficients of the expression (A) are respectively divisible by \( (x-a)^m, (x-a)^{m-1} \), \( (x-a)^{m-2} \), &c.; and, consequently, we shall have

\[ X = 0, \quad Y = 0, \quad Z = 0, \quad V = 0, \quad \text{&c.} \]

when the common root \( a \) is substituted for \( x \).

If the polynome \( X \) is divisible by \( \left\{ (x-\omega)^2 + \beta^2 \right\}^n \), it may be proved in like manner, that \( \left\{ (x-a)^2 + \beta^2 \right\}^{n-1} \) will be a common divisor of X and Y.

We may, therefore, lay down the following rule, for finding all the double, triple, &c. divisors of any given polynome X: "Find R, the greatest common measure of X and Y, and resolve it into its elementary factors; then each of these factors will be contained in X once more than in R."

7. If it be required to find how many of the roots of an equation are positive, and how many are negative, we have for this purpose the rule first published in the Geometry of Descartes. This celebrated rule seems to have been discovered by induction; at least, its author gave no demonstration of it, and disputes arose about its true import. It was demonstrated for the first time by Du Gua, in the Mémoires de Paris; but many other demonstrations of it have since appeared, of which that of Segner, in the Mémoires de Berlin 1756, is not only the most simple, but probably the most simple that will ever be invented.

Segner deduced the rule of Descartes from the following analytical proposition, viz.

"If any rational polynome be multiplied by \( x-a \), the changes from one sign to another, from + to -1, and from - to +1, will be at least one more in the product, than in the given polynome; and if it be multiplied by \( x+a \), the successions of the same sign, of + to +, and of - to -1, will be at least one more."

Let the proposed polynome be

\[ x^n \pm A x^{n-1} \pm B x^{n-1} \ldots \pm M x \pm N; \]

then, according to the usual process, the product of the polynome by \( x-a \) will be found by adding these two lines, viz.

\[ \begin{aligned} &x^{n+1} \pm A x^n \pm B x^{n-1} \ldots \pm M x^2 \pm N x \\ &-a x^n \mp A a x^{n-1} \mp L a x^2 \mp M a x \mp N a \end{aligned} \]

the signs of the several terms remaining unchanged in the first line, and being all changed in the second line. It is evident, therefore, that the terms of the product will have the same signs with the respective terms of the proposed polynome, except when a coefficient in the second line is greater than the one above it, and likewise has a contrary sign; the sign of the last term of the product being always the same with the sign of the last term of the second line. Now, beginning on the left hand, pass over the terms of the first line, so long as they have the same signs with the terms of the product. When this ceases to be the case, the signs in the product will be the same as in the second line, and contrary to those in the first line; wherefore descend to the second line, and pass along its terms till the signs in the product are again the same as those in the first line, and then ascend to that line. Continue thus descending and ascending alternately, till all the terms in both lines are taken in. At the conclusion, it is evident, that the descensions are always one more than the ascensions, because the passing from one line to another both begins and ends with descending.

If we descend from \( \pm A x^n \), in the first line, to \( \pm A a x^{n-1} \), in the second line, it is evident, that the signs of \( \pm A x^n \) and \( \pm B x^{n-1} \), in the first line, will be the same, both being contrary to the sign of \( \pm A a x^{n-1} \), in the second line. Therefore, in the given polynome, the first and second terms have the same sign. But in the product, the like terms have contrary signs; for the second term of the product has the same sign with \( \pm A x^n \) in the first line, and the third term of the product has the same sign with \( \pm A a x^{n-1} \) in the second line. Thus, it appears that a variation, from one sign to another, is introduced in the product, instead of a continuation of the same sign that takes place in the given polynome; and the same thing will happen at every descending.

In ascending from the second line to the first, there may either be a continuation in the product instead of a variation in the given polynome, or the contrary: but one of these two must take place.

Now, so long as we keep on the first line, the signs in the product are the same with those of the given polynome; and, so long as we keep on the second line, the signs in the product are contrary to those in the polynome. In both cases, therefore, the variations from + to -1, and from - to +1, are the same in the product and in the polynome. Every descending introduces a variation in the product, instead of a continuation that takes place in the polynome; and although it be supposed that every ascending introduces a continuation in the product instead of a variation that exists in the polynome, yet, on the whole, the variations introduced must be one more than the continuations, because the descensions are one more than the ascendings.

Again, if the given polynome be multiplied by \( x+a \), the product will be the sum of these two lines, viz.

\[ \begin{aligned} &x^n \pm A x^n \pm B x^{n-1} \ldots \pm M x^2 \pm N x \\ &+ a x^n \pm A a x^{n-1} \pm L a x^2 \pm M a x \pm N a \end{aligned} \]

Here the terms of both lines have the same signs; and, as before, the signs in the product will be the same with the signs of the proposed polynome, unless when a coefficient in the second line is greater than the one above it, and likewise has a contrary sign; the sign of the last term of the product being always the same with the sign of the last term in the second line. Now, if we pass along all the terms of both lines, descending from the first line to the second, when the signs in the product change from being the same with those in the given polynome, to be contrary to them; and ascending from the second line to the first, when the signs in the product change from being contrary to those in the polynome, to be the same with them; it is evident, that the descensions will be one more than the ascendings, as in the former case.

If we descend from \( \pm A x^n \) in the first line, to \( \pm A a x^{n-1} \) in the second line, the two terms \( \pm A x^n \) and \( \pm B x^{n-1} \) in the first line, will have different signs; for, on account of the descending, \( \pm B x^{n-1} \) has a contrary sign to the term \( \pm A a x^{n-1} \) below it, and, consequently, to \( \pm A x^n \) in the first line. Therefore the second and third terms in the polynome have different signs. But the like terms in the product have the same sign; for the second term in the product has the same sign with \( \pm A x^n \) in the first line; and the third term of the product has the same sign with \( \pm A a x^{n-1} \) in the second line. Thus there is a continuation of the same sign introduced in the product, instead of a variation from one sign to another that takes place in the polynome; and the same thing is true at every descending.

In ascending from the second line to the first, there may either be a variation in the product instead of a continuation that exists in the polynome, or the contrary. But one of these two must take place.

Now, it is evident, that, except at the descendencies and ascendings, there is the same number of continuations of the same sign, and the same number of variations from one sign to another, in the product and in the given polynome. Every descending introduces a continuation in the product instead of a variation existing in the polynome. And even if we suppose that every ascending introduces a variation in the product instead of a continuation that takes place in the polynome, yet, on the whole, there will be one continuation more in the product than in the polynome, because the descendencies are one more than the ascendings.

In the preceding demonstration it is supposed, that all the ascendings have a contrary effect to the descendencies, by which means there is introduced in the product the least possible number of variations from one sign to another in the one case, and the least possible number of continuations of the same sign in the other. But if, in the first case, we suppose that, at one ascending, there is a variation in the product, and a continuation in the polynome, this will add one to the variations in the product, and one to the continuations in the polynome; so that, the variations in the product will now exceed those in the polynome by three, namely, by two more than in the circumstances supposed in the demonstration. And if we extend the like reasoning to two, three, &c. ascendings, the variations in the product will exceed those in the polynome, respectively by five, seven, &c. The like conclusion is evidently true of the second case, mutatis mutandis; and hence the preceding proposition, when it is generalized as much as it can be, may be thus enunciated: "If any rational polynome be multiplied by \( x-a \), the variations from one sign to another in the product will exceed those in the polynome by one, or three, or five, or by some odd number; and if it be multiplied by \( x+a \), the continuations of the same sign in the product will exceed those in the polynome by one, or three, or five, or by some odd number."

Now, if we conceive that any rational polynome is resolved into its binomial factors; there will be a factor of the form \( x-a \) for every positive root, and one of the form \( x+a \) for every negative root; and when all the factors are multiplied together in order to reproduce the polynome, it follows, from what has been proved, that the product will contain at least one change from \( + \) to \( - \), or from \( - \) to \( + \), for every factor of the form \( x-a \), or for every positive root; and at least one succession of \( + \) to \( + \), or of \( - \) to \( - \), for every factor of the form \( x+a \), or for every negative root. Hence this rule, viz. "An equation cannot have more positive roots than it has variations from one sign to another; nor more negative roots than it has continuations of the same sign."

In general, this rule merely points out limits which the number of the positive and negative roots of an equation cannot exceed. But it gives no criterion by which we can certainly know that an equation has even one positive or one negative root, much less does it ascertain the exact number of each kind.

But if the proposed equation can be completely resolved into real binomial factors; in which case the total number of its roots will be equal to its dimensions, and, consequently, to the sum of all the variations from one sign to another, and of all the continuations of the same sign; it is evident, that the number of the positive roots will be precisely equal to that of the variations, and the number of the negative roots precisely equal to that of the continuations. In this case, therefore, and in this case only, the rule of Descartes is perfect, ascertaining the exact number of each kind of roots in the proposed equation.

We subjoin some consequences that result from the principles laid down.

"If a polynome \( f(x) \) of \( n \) dimensions be multiplied by \( x-a \), or \( x+a \); and, in the first case, if the number of variations from one sign to another be augmented by the odd number \( 2i+1 \); or, in the second case, if the number of continuations of the same sign be augmented by \( 2i+1 \); then the total number of the roots, positive and negative, of the proposed polynome, cannot be greater than \( n-2i \)."

For, when the multiplier is \( x-a \), let \( m \) denote the number of the variations from one sign to another, in the proposed polynome \( f(x) \); then \( m+2i+1 \) will be the total number of variations in the product \( (x-a) \times f(x) \): consequently, the total number of continuations in \( (x-a) \times f(x) \) will be equal to \( (n+1)-(m+2i+1) \), or \( n-m-2i \). But a polynome cannot have more negative roots than it has continuations of the same sign; wherefore, the number of the negative roots of \( (x-a) \times f(x) \) cannot be greater than \( n-m-2i \). Now, the two polynomes \( f(x) \) and \( (x-a) \times f(x) \) have the same negative roots; and hence the number of the negative roots of \( f(x) \) cannot exceed \( n-m-2i \). But the number of the positive roots of \( f(x) \) cannot exceed \( m \); consequently, the total number of the roots of \( f(x) \) cannot be greater than \( m+n-m-2i \); that is, than \( n-2i \). And the proposition may be demonstrated in a similar manner when the multiplier is \( x+a \).

"If one, or several consecutive terms, of an equa- tion be wanting; and, if the next terms on each side of those wanting have the same sign, the equation cannot have as many roots as it has dimensions."

Let the equation be \( P + Q = 0 \), \( P \) and \( Q \) denoting the two parts on each side of the terms wanting. Having multiplied \( P + Q \) by \( x - a \), the product will be \( (x - a)P + (x - a)Q \); and it is evident that we may consider \( P, Q, (x - a)P, (x - a)Q \) as separate polynomials; hence, in each of the polynomials \( (x - a)P \) and \( (x - a)Q \) there will be at least one more variation from one sign to another, than there is in \( P \) and \( Q \). Again, in the polynomial \( P + Q \), there will be a continuation of the same sign, in passing from \( P \) to \( Q \); because the last term of \( P \) is supposed to have the same sign with the first term of \( Q \). On the other hand, because the last term of \( (x - a)P \) has a contrary sign to the last term of \( P \); and the first term of \( (x - a)Q \), the same sign with the first term of \( Q \), it follows that, in the polynomial \( (x - a)P + (x - a)Q \), there will be a variation from one sign to another, in passing from \( (x - a)P \) to \( (x - a)Q \). Therefore, on the whole, there will be at least three variations from one sign to another in \( (x - a)P + (x - a)Q \), more than there is in \( P + Q \): Consequently, by the last proposition, the number of all the roots of the proposed equation must be at least two less than its dimensions.

8. An important inquiry is, to find how many roots, that is, real roots, there are in any proposed equation. Much has been written on this subject, but not very successfully. No general method has been found that is practically useful. Many criteria have been contrived, by means of which we can certainly discover that roots are wanting in an equation; although we cannot infer the existence of the roots when the same criteria fail. But great value cannot be attached to such rules; since they are neither sufficient guides in practice, nor have much tendency to throw light on the theory.

Waring first, and nearly about the same time Lagrange, proposed a method which is successful in finding the conditions necessary in order that an equation have as many roots as it has dimensions; and which, in all cases, points out a limit that the number of the roots cannot exceed. This is effected by an auxiliary equation, and merely by the signs of its coefficients, without requiring the computation of any of its roots. This procedure answers very well for equations of the third and fourth degrees; and it has even been extended by Waring to those of the fifth degree; but, in this last case, the calculation is very long, and would be altogether impracticable in the higher orders of equations. It is also not a little probable that this rule employs more conditions than are absolutely necessary for determining the point in question; there being great reason to think that some of them are implied in the rest, and are deducible from them. The method here alluded to depends upon the theory of trinomial divisors; and, as it is much referred to by algebraists of the present day, we shall, in a subsequent part of this article, briefly explain the principles on which it is founded.

There is also another way of finding the number of real roots in an equation, which is general for all orders, and requires the solution of such equations only as are of lower dimensions than the one proposed. As to practical utility, indeed, this method is of little avail in equations passing the third and fourth degrees, or, at most, the fifth degree; but it is, nevertheless, not without interest; both because it is founded on the principles essential to the inquiry, and because it leads to some useful properties. Algebraists differ from one another in their exposition of this method. Some derive it from the theory of Harriot, namely, that every rational polynomial is the product of as many binomial factors as it has dimensions; in which manner of proceeding the impossible roots are the occasion of uncertainty and embarrassment. Others, again, deduce it from the variations of magnitude which a rational polynomial undergoes when the unknown quantity is made to pass through all possible degrees of increasing and decreasing. This last mode of investigation seems greatly to deserve the preference, being in reality the only one that is entirely unexceptionable, and requires no principles foreign to the research.

Suppose an equation, \( x^n + A x^{n-1} + B x^{n-2} \ldots + M x + N = 0 \), which we may denote by \( f(x) = 0 \): substitute \( x - i \) in place of \( x \), and put

\[ X = f(x-i) = x^n + A x^{n-1} \ldots + M x + N, \] \[ X' = n x^{n-1} + (n-1) A x^{n-2} + (n-2) B x^{n-3} \ldots + M, \] \[ X'' = n(n-1) x^{n-2} + n(n-1) \frac{n-2}{2} A x^{n-3} + &c. &c. \]

then the function \( f(x-i) \) will be transformed into

\[ X - X'. i + X''. i^2 - X''''. i^3 + &c. \]

If we use the notation of the differential calculus, the same transformation will be thus represented,

\[ f(x) - \frac{d f(x)}{dx} i + \frac{1}{2} \frac{d^2 f(x)}{dx^2} i^2 - &c.; \]

which has the advantage of pointing out in what manner the several functions, \( X', X'', &c. \) are derived from one another, and from the first function \( X \), or \( f(x) \).

Let \( \alpha, \beta, \gamma, &c. \) denote the real roots of the equation \( X = 0 \), or \( f(x) = 0 \), arranged according to the order of their magnitude, that is, \( \alpha \) greater than \( \beta \), \( \beta \) greater than \( \gamma \), and so on. In like manner, observing the same order of arrangement, let \( \alpha', \beta', \gamma', &c. \) represent the roots of \( X' = 0 \), or \( \frac{d f(x)}{dx} = 0 \); and for the sake of simplicity, suppose that the equation \( X = 0 \) has no equal roots.

The relations, which the variations of the polynomial \( X \) bear to the variations of \( x \), depend upon the functions \( X', X'', &c. \) and principally upon the first of these. If \( X' \) be positive, \( X \) will decrease as \( x \) decreases; if \( X' \) be negative, \( X \) will increase as \( x \) decreases; and if \( X' \) pass from being positive to become negative, or the contrary, then \( x \) continuing to decrease, \( X \) will change from decreasing to increasing, or the contrary; that is, it will attain a minimum or a maximum value. What is here said is the foundation of the method taught in the differential calculus, for finding the maxima and minima of algebraic quantities.

Now, when \( x \) has a value great enough, the poly- nome X' will have the same sign with its first term, that is, it will be positive; and it will continue positive so long as x is greater than \( \alpha' \), the greatest root of the equation \( X'=0 \); after which it will become negative. Hence, while x decreases to the limit \( \alpha' \), the polynome \( f(x) \), which is positive when x is sufficiently great, will continually decrease; and when \( x=\alpha' \), \( f(x) \) will pass from decreasing to increasing, or it will have a minimum value. Now, if this minimum \( f(\alpha') \) be positive, \( f(x) \) has not decreased to zero, and the given equation will have no root greater than \( \alpha' \). If \( f(\alpha')=0 \), then, because the two equations, \( X=0 \) and \( X'=0 \), take place at the same time, the given equation will have two roots equal to \( \alpha' \). (Sect. 6.) Lastly, if \( f(\alpha') \) be negative, the polynome \( f(x) \) has decreased from being positive to be negative; and therefore it has passed through zero, and the given equation will have one root, viz. \( \alpha \) greater than \( \alpha' \).

As x continues to decrease from \( \alpha' \) to \( \beta' \), the polynome X' being negative, \( f(x) \) will continually increase. At the limit \( x=\beta' \), X' is first equal to zero, and then becomes positive; and \( f(x) \) will therefore change from increasing to decreasing, or will attain a maximum value. If this maximum \( f(\beta') \) be negative, the polynome \( f(x) \) has not increased to zero, and the given equation will have no root between \( \alpha' \) and \( \beta' \): if \( f(\beta')=0 \), it will have two roots equal to \( \beta' \); and if \( f(\beta') \) be positive, \( f(x) \), in increasing from the negative quantity \( f(\alpha') \) to the positive quantity \( f(\beta') \), must have passed through zero, and the given equation will have one root, viz. \( \beta \), between \( \alpha' \) and \( \beta' \).

In like manner, x continuing to decrease from \( \beta' \) to \( \gamma' \), the polynome \( f(x) \) will decrease from the maximum \( f(\beta') \) to the minimum \( f(\gamma') \): if \( f(\gamma') \) be positive, the proposed equation will have no root between \( \beta' \) and \( \gamma' \); if \( f(\gamma')=0 \), it will have two roots equal to \( \gamma' \); and if \( f(\gamma') \) be negative, it will have one root, viz. \( \gamma \), between the limits \( \beta' \) and \( \gamma' \).

As the function \( f(x) \) must become a minimum or a maximum, or must pass from decreasing to increasing, or the contrary, between every two contiguous roots of the equation \( f(x)=0 \); and as the limits where the changes take place are determined by the roots of the equation \( X'=0 \); it follows that there must be at least one root of this last equation between every two contiguous roots of the first. Hence the equation \( f(x)=0 \) cannot have as many roots as dimensions, unless the equation \( X'=0 \) likewise have as many roots as dimensions; and, in general, we have this rule, which determines a limit that the number of the roots of an equation cannot surpass, although it may fall short of it: "The roots of an equation \( f(x)=0 \) cannot exceed in number those of the equation \( \frac{d f(x)}{dx}=0 \), by more than one."

But if we can find the roots of the equation \( X'=0 \), which is always one degree lower than the proposed equation, we can thence discover exactly both the number and the limits of the roots of this last. For let \( \alpha', \beta', \gamma', \) &c. be substituted in the polynome \( f(x) \), and let the results be arranged in order, viz.

\[ f(\alpha'), f(\beta'), f(\gamma'), f(\delta'), \text{ &c.}: \]

if these quantities are alternately negative and positive; the first, third, fifth, &c. which are all minima, having the sign minus; and the second, fourth, &c. which are all maxima, having the sign plus; then the proposed equation \( f(x)=0 \) will have just one root more than the equation \( X'=0 \). When some of the conditions fail, the roots of the proposed equation will fall short of the numbers specified. If one maximum have the sign minus, or one minimum the sign plus, two roots will be wanting in the proposed equation; and, in general, as many roots will disappear, as there are consecutive minima and maxima that have the same sign deducting one; unless the minima and maxima precede the greatest root, or come after the least root, in which cases there will be as many roots wanting as there are minima and maxima that have the same sign.

Since the series of functions, X, X', X", &c. are derived similarly from one another, we may prove, as has been done with respect to the two first, that the roots of any one are contained between the roots of that which follows it. Hence, if the given equation have as many roots as dimensions, every equation in the series will likewise have as many roots as dimensions; and if there be roots wanting in any one, there will be at least as many wanting in every equation preceding it in the series.

The connected equations necessarily terminate in one of the first degree, which gives a limit between the two roots of the quadratic immediately before it; in like manner, the roots of the quadratic are the limits of the roots of the cubic preceding it; and, in this manner, by going through all the successive equations, we shall finally arrive at the limits of the roots of the proposed equation. This process has been called La Methode des Cascades; but the length of the calculations render it useless in practice.

The procedure explained above would enable us to find the number of roots in an equation of any order, if we were in possession of rules for solving equations of the inferior degrees. For want of such rules, the practical advantage that can be derived from it is very limited. Mathematicians have, therefore, turned their attention to determine the point in question in a way that should not require the resolution of equations. They have sought to investigate rational functions of the coefficients, which, by means of the signs they are affected with in every particular case, might indicate the number of roots the equation possesses. Of this nature is the method which Du Gua has given in the Memoires de Paris, 1741, for finding the conditions necessary in order that an equation have as many roots as dimensions. By a process analogous to that of Du Gua, M. Cauchy, in an excellent Memoir, published in the sixteenth volume of the Journal de l'Ecole Polytechnique, has shown not only that the total number of the roots may, in every case, be discovered, but likewise, that the numbers of the positive and negative roots may be separately ascertained. The principles of both these methods are to be found in the theory explained above; but, as many considerations of some intricacy are involved in them, a particular account of them would exceed the limits of this article. In what goes before, we have supposed that all the roots of the equation \( X' = 0 \) are unequal; and, in order to complete the theory, it remains to notice the consequences that follow when the case is otherwise.

Suppose, then, that \( X' = (x - \lambda)^i \times Q \): And, in the first place, if \( \lambda \) be a root of the equation \( f(x) = 0 \), there will, in reality, be no exception to the general conclusion; because, in this case, it is known that the polynome \( f(x) \) will be divisible by \( (x - \lambda)^{i+1} \). (Sect. 6.) Now, the case just mentioned being set aside, if \( i \) be an even number, the polynome \( X' \), or \( (x - \lambda)^i \cdot Q \), will be equal to zero when \( x = \lambda \); but it will not change its sign when \( x \), from being less, comes to be greater than \( \lambda \). Hence the polynome \( f(x) \) will neither attain a maximum nor a minimum value at the same limit; and it will have no root, either between \( \lambda \) and the next greater root of the equation \( X' = 0 \), or between \( \lambda \) and the next less root of the same equation. It appears, therefore, that, when \( i \) is even, the number of the roots of the equation \( f(x) = 0 \), and their limits, will depend entirely upon the equation \( Q = 0 \). Again, when \( i \) is an odd number, the polynome \( X' \) will be equal to zero when \( x = \lambda \), and it will likewise change its sign when \( x \) is taken on contrary sides of that limit: Consequently, when \( x = \lambda \), the polynome \( f(x) \) will be a maximum or a minimum; and the nature of its roots will depend upon the equation \( (x - \lambda)Q = 0 \). It is evident that we may extend the same conclusions to any two adjacent equations in the series, \[ X = 0, X' = 0, X'' = 0, X''' = 0, \text{ &c.} \] provided the one which stands lower in the series is reducible to the form \( (x - \lambda)^i Q \); and that \( x - \lambda \) is not a common divisor of both. We may likewise draw this general inference from the principles that have been explained, viz. "If, in the series of connected equations, any one be found which is divisible by \( (x - \lambda)^{2i} \), or \( (x - \lambda)^{2i+1} \), at the same time that \( x - \lambda \) is not a divisor of the equation immediately preceding, there will be at least \( 2i \) roots wanting in this last equation, and in all that stand before it in the series."

The following not inelegant proposition is a consequence of what has just been proved: "The number of the roots of an equation of \( n \) dimensions, in which \( 2i \) or \( 2i + 1 \), consecutive terms, are wanting, cannot be greater than \( n - 2i \)."

Let the equation be represented by \[ P + Q = 0; \] supposing that \( 2i \), or \( 2i + 1 \) terms, are wanting between P and Q. Therefore, if the first term of Q contain \( x^m \), the last term of P will contain \( x^{m+2i+1} \), or \( x^{m+2i+2} \). Now, in the series of equations, we shall at length arrive at one from which all the quantities of Q are exterminated; which equation, if we use the notation of the Differential Calculus, is equivalent to \[ \frac{d^{m+1}P}{dx^{m+1}} = 0; \] and it is divisible by \( x^{2i} \), or \( x^{2i+1} \). And, as the one immediately preceding it in the series, viz. \[ \frac{d^mP}{dx^m} + \frac{d^mQ}{dx^m} = 0, \] is not divisible by \( x \), it follows from what has been shown, that there will be at least \( 2i \) roots wanting in this last equation, and in all those that stand before it; consequently, the proposed equation cannot have more than \( n - 2i \) roots.

From this we learn, that it is not always possible, at least by any operations with real quantities, to transform an equation into another in which any proposed number of the intermediate terms shall be wanting. For the terms to be taken away may be such, that the transformed equation could not have the same number of real roots as the one given; but it is impossible, without introducing imaginary quantities, to transform an equation with a certain number of real roots into another with a different number of such roots.

9. In what goes before, we have sought for the roots and binomial divisors in the nature of the polynome. We are now to take an inverted view of the subject, and to consider a rational polynome as produced by the continued multiplication of as many binomial factors as it has dimensions; from which source there arises an interesting set of properties.

If we take the words, root and binomial factor, strictly in the sense in which we have hitherto used them, and as denoting real quantities only, nothing is more certain than that all polynomes cannot be generated by binomial factors. But it will afterwards be proved, that every rational polynome can be completely exhausted by binomial and trinomial divisors; and if we admit the resolution of every trinomial divisor into two imaginary factors, we shall arrive, with all the rigour of which the investigation is capable, at the genesis of equations supposed by Harriot, which represents them as entirely composed of binomial factors, possible or impossible. Besides, in extending to all equations the conclusions obtained from the manner of generating them, it may be observed, that the properties so obtained, being ultimately expressed in functions of the coefficients from which the roots and generating factors have disappeared, are in a manner independent of the method of investigation. Such is the structure of the language of algebra, that the conclusions to which it leads, although deduced by reasoning from a hypothesis not strictly general, are nevertheless true in all cases, when they are finally disengaged from what is peculiar in the analysis.

Suppose a polynome, as \[ x^n - A^{(1)} x^{n-1} + A^{(2)} x^{n-3} \ldots \pm A^{(n-1)} x + A^{(n)}, \] which is produced by the multiplication of the \( n \) factors, \[ (x - \alpha) \cdot (x - \beta)(x - \gamma)(x - \delta) \text{ &c.}: \] then, by actually multiplying the factors, and equating the like terms of the equivalent expressions, we shall get \[ A^{(1)} = \alpha + \beta + \gamma + \delta + \text{&c.} \] \[ A^{(2)} = \alpha \beta + \alpha \gamma + \text{&c.} + \beta \gamma + \text{&c.} \] A^{(3)} = \alpha \beta \gamma + \alpha \beta \delta + \&c. + \beta \gamma \delta + \&c.

A^{(4)} = \alpha \beta \gamma \delta + \&c.

\&c.

Hence, it appears that the coefficient of the second term of the polynome, or —A^{(1)}, is equal to the sum of all the roots with their signs changed; the coefficient of the third term, or +A^{(2)}, to the sum of all the products of every two roots; the coefficient of the fourth term, or —A^{(3)}, to the sum of all the products of every three roots with their signs changed, and so on, the signs of the roots being always changed in the products of an odd number; and, finally, the last term is the product of all the roots with their signs changed or not, according as their number is odd or even.

It is evident, that the ultimate product of the binomial factors will always be the same, in whatever order they are multiplied; and hence the coefficients of the polynome will consist of the same products, however the roots be interchanged among one another. Expressions of the kind just mentioned, which have constantly the same value, whatever change is made in the order of the quantities they contain, are called invariable functions and symmetrical functions. The coefficients of an equation are the most simple symmetrical functions of the roots, from which it may be required, on the one hand, to deduce all other functions of the like kind, and, on the other, to go back to the roots themselves. Most inquiries relating to equations are connected with one or other of these two problems; of which the first, like most direct methods, is attended with little difficulty, and has been completely solved; while the other, past equations of the fourth degree, has eluded all the attempts of algebraists.

After the coefficients of the polynome, the next most simple symmetrical functions of the roots are the sums of the squares, cubes, &c. In the universal arithmetic of Sir Isaac Newton, a very elegant rule is given for computing the sum of any proposed powers of the roots; and as this rule is a fundamental point in the theory of equations, we subjoin an elementary investigation of it.

Of the binomial factors before set down, let the first x—α be left out, and, having multiplied the rest together, let the product be,

x^{n-1} - \varphi^{(1)} x^{n-2} + \varphi^{(2)} x^{n-3} - \varphi^{(3)} x^{n-4} + \&c.;

in which expression \( \varphi^{(1)} \) is the sum of all the roots \( \beta, \gamma, \delta, \&c. \) except the first \( \alpha \); \( \varphi^{(2)} \) is the sum of the products of every two of them, and so on. Now, multiply by \( x-\alpha \), and the product will be equivalent to the given polynome: hence we get

\[ \begin{align*} A^{(1)} &= \alpha + \varphi^{(1)} \\ A^{(2)} &= \alpha \cdot \varphi^{(1)} + \varphi^{(2)} \\ A^{(3)} &= \alpha \cdot \varphi^{(2)} + \varphi^{(3)} \\ \vdots \\ A^{(r)} &= \alpha \cdot \varphi^{(r-1)} + \varphi^{(r)} \end{align*} \]

Again, multiply these formulae in order by \( \alpha^{r-1}, \alpha^{r-2}, \alpha^{r-3}, \&c. \); then

\[ \begin{align*} A^{(1)} \cdot \alpha^{r-1} &= \alpha^r + \alpha^{r-1} \cdot \varphi^{(1)} \\ A^{(2)} \cdot \alpha^{r-2} &= \alpha^{r-1} \cdot \varphi^{(1)} + \alpha^{r-2} \cdot \varphi^{(2)} \\ \vdots \\ A^{(r-1)} \cdot \alpha &= \alpha^2 \varphi^{(r-2)} + \alpha \cdot \varphi^{(r-1)} \\ A^{(r)} &= \alpha \cdot \varphi^{(r-1)} + \varphi^{(r)} \end{align*} \]

and, by adding and subtracting alternately, we get

\[ \alpha^{(r)} - A^{(1)} \cdot \alpha^{r-1} + A^{(2)} \cdot \alpha^{r-2} \ldots \pm A^{(r-1)} \cdot \alpha \] \[ = \mp A^{(r)} = \mp \varphi^{(r)}, \]

in which expression \( \varphi^{(r)} \) is the sum of all the products of \( r \) dimensions of the roots \( \beta, \gamma, \delta, \&c. \) leaving out the first \( \alpha \).

In like manner, if we leave out the factor \( x-\beta \), and multiply all the rest, and proceed as before, we shall get

\[ \beta^r - A^{(1)} \beta^{r-1} + A^{(2)} \beta^{r-2} \ldots \pm A^{(r-1)} \beta \mp A^{(r)} \] \[ = \mp \varphi^{(r)}, \]

the symbol \( \varphi^{(r)} \) being the sum of the products of \( r \) dimensions of all the roots \( \alpha, \gamma, \delta \), except the second \( \beta \).

And, if we next leave out the factor \( x-\gamma \), and follow a like procedure, we shall get

\[ \gamma^r - A^{(1)} \gamma^{r-1} + A^{(2)} \gamma^{r-2} \ldots \pm A^{(r-1)} \gamma \mp A^{(r)} \] \[ = \mp \varphi^{''(r)}; \]

where \( \varphi^{''(r)} \) represents the sum of the products of \( r \) dimensions of all the roots \( \alpha, \beta, \delta, \&c. \) except the third \( \gamma \).

If we proceed similarly till every one of the \( n \) factors is left out in its turn, and then add all the results, we shall get

\[ S_r - A^{(1)} S_{r-1} + A^{(2)} S_{r-2} \ldots \pm A^{(r-1)} S_1 \] \[ = \mp n A^{(r)} = \mp \left\{ \varphi^{(r)} + \varphi^{(r)} + \varphi^{''(r)} + \&c. \right\}; \]

in which expression \( S_r \) is written for the sum of the \( r \) powers of the roots; \( S_{r-1} \), for the sum of the \( (r-1) \) powers, and so on.

Every product in any one of the aggregate quantities, \( \varphi^{(r)}, \varphi^{(r)}, \varphi^{''(r)}, \&c. \), is found in \( A^{(r)} \), which is the sum of the products of \( r \) dimensions of all the roots: and, hence, it is easy to perceive that the sum of all the aggregates must be a multiple of \( A^{(r)} \). Take any product in \( A^{(r)} \): then that product will not be contained in \( r \) of the quantities \( \varphi^{(r)}, \varphi^{(r)}, \varphi^{''(r)}, \&c. \); because, in so many of them, one or other of the letters of the product Equations. will be wanting; but the same product will be contained once in every one of the \( n-r \) remaining quantities, because, in every one of these, all the letters of the product will be contained. Every product in \( A^{(r)} \) is, therefore, repeated \( n-r \) times in the sum of the quantities \( \varphi^{(r)}, \varphi^{(r)}, \varphi^{(r)}, \) &c.: consequently, \[ \varphi^{(r)} + \varphi^{(r)} + \varphi^{(r)} + \text{&c.} = (n-r)A^{(r)}. \] Substitute this value in the formula obtained above, and, after transposing and cancelling \( nA^{(r)} \), which appears with contrary signs, we shall get \[ S_r - A^{(1)}S_{r-1} + A^{(2)}S_{r-2} \ldots \pm A^{r-1}S_1 = 0. \] This is the rule of Sir Isaac Newton, and contains all his particular formulae, as will readily appear by putting 1, 2, 3, &c. successively for \( r \).

The preceding formula will enable us to compute, in succession, the sums of all the positive powers of the roots, both when \( r \) is less, and when it is greater than the dimensions of the equation. But, in applying the formula in the latter case, we must observe that all the coefficients of the polynome after \( A^{(r)} \) are wanting, or equal to nothing.

If, in the first step of the preceding investigation, we take the coefficients that follow \( A^{(r)} \), we shall get \[ \begin{align*} A^{(r+1)} &= \alpha \cdot \varphi^{(r)} + \varphi^{(r+1)} \\ A^{(r+2)} &= \alpha \cdot \varphi^{(r+1)} + \varphi^{(r+2)} \\ &\vdots \\ A^{(n-1)} &= \alpha \varphi^{(n-2)} + \varphi^{(n-1)} \\ A^{(n)} &= \alpha \varphi^{(n-1)} \end{align*} \] And, by first dividing by \( \alpha, \alpha^2, \alpha^3, \) &c. in order, and then subtracting and adding alternately, we shall obtain \[ \frac{A^{(r+1)}}{\alpha} - \frac{A^{(r+2)}}{\alpha^2} + \frac{A^{(r+3)}}{\alpha^3} - \text{&c.} = \varphi^{(r)}. \] In a similar manner, we get \[ \frac{A^{(r+1)}}{\beta} - \frac{A^{(r+2)}}{\beta^2} + \frac{A^{(r+3)}}{\beta^3} - \text{&c.} = \varphi^{(r)} \] \[ \frac{A^{(r+1)}}{\gamma} - \frac{A^{(r+2)}}{\gamma^2} + \frac{A^{(r+3)}}{\gamma^3} - \text{&c.} = \varphi^{(r)} \] Therefore, by adding all these formulae, and substituting for the sum of \( \varphi(r), \varphi^{(r)}, \) &c. the value of it already found, we shall finally obtain \[ A^{(r+1)}S_{-1} - A^{(r+2)}S_{-2} + A^{(r+3)}S_{-3} - \text{&c.} \] \[ = (n-r)A^{(r)}, \] the symbols \( S_{-1}, S_{-2}, \) &c. being put for the sums of the negative powers of the roots according to the indices underwritten. This formula will enable us to compute the sums of the negative powers of the roots.

If, in the formula for the sums of the positive powers of the roots, we make \( r \) successively equal to 1, 2, 3, &c. we shall get \[ \begin{align*} A^{(1)} &= S_1 \\ -2A^{(2)} &= -A^{(1)}S_1 + S_2 \\ 3A^{(3)} &= A^{(2)}S_1 - A^{(1)}S_2 + S_3 \\ -4A^{(4)} &= -A^{(3)}S_1 + A^{(2)}S_2 - A^{(1)}S_3 + S_4 \\ &\text{&c.} \end{align*} \] and from this we learn that the quantities \( S_1, S_2, S_3, \) &c. may be found by means of this expression, viz. \[ \frac{A^{(1)} - 2A^{(2)}z + 3A^{(3)}z^2 \ldots \pm A^{(n)}z^{n-1}}{1 - A^{(1)}z + A^{(2)}z^2 \ldots \pm A^{(n)}z^n} = S_1 + S_2z + S_3z^2 + \text{&c.} \] for if we multiply the series on the right-hand side of the sign of equality, by the denominator of the fraction on the other side, and then equate the coefficients of the product to the like coefficients of the numerator, we shall obtain the very formulae set down above. Hence the sums of the powers of the roots expressed in terms of the coefficients of the polynome, will be found by developing the fraction in a series. In effecting the developement different analytical methods may be followed; and the quantities sought will thus be obtained by different rules, or exhibited in expressions of different forms, such as those given by Waring, Vandermonde, Euler, and La Grange.

And in like manner if, in the formula for the sums of the negative powers of the roots, we make \( r \) successively equal to \( n-1, n-2, n-3, \) &c. we shall get \[ \begin{align*} A^{(n-1)} &= A^{(n)}S_{-1} \\ -2A^{(n-2)} &= -A^{(n-1)}S_{-1} + A^{(n)}S_{-2} \\ 3A^{(n-3)} &= A^{(n-2)}S_{-1} - A^{(n-1)}S_{-2} + A^{(n)}S_{-3} \\ -4A^{(n-4)} &= -A^{(n-3)}S_{-1} + A^{(n-2)}S_{-2} \\ &\text{&c.} \end{align*} \] from which it appears that the values of all the quantities \( S_{-1}, S_{-2}, S_{-3}, \) &c. will be obtained by means of this expression, viz. \[ \frac{A^{(n-1)} - 2A^{(n-2)}z + 3A^{(n-3)}z^2 - \text{&c.}}{A^{(n)} - A^{(n-1)}z + A^{(n-2)}z^2 - \text{&c.}} = S_{-1} \] \[ +S_{-2}z + S_{-3}z^2 + \text{&c.} \]

Two kinds of quantities only can enter into any rational and symmetrical function of the roots of an equation; and these are, the sums of the like powers of the roots, and the sums of such products as, \( \alpha_i \beta_j \gamma^m \) &c. which arise from multiplying different powers of the roots, two and two, three and three, &c. We shall now shortly point out in what manner the latter sums are deduced from the sums of the like powers, for the computation of which rules have already been given; by which means we shall be enabled to find the value of any proposed function of the kind above mentioned. Let it be required to find the sum of all the products, such as \( \alpha^i \beta^{j'} \), that arise from combining two powers of the roots in all possible ways; which sum may be denoted by the symbol \( \Sigma \alpha^i \beta^{j'} \). Now it is evident that the product, \( S_i \times S_{j'} \), will contain two sorts of terms only, namely, powers of the roots, such as \( \alpha^{i+j'} \), and the products of which the sum is sought; therefore

\[ \Sigma \alpha^i \beta^{j'} = S_i \times S_{j'} - S_{i+j'} \]

Next let it be required to find \( \Sigma \alpha^i \beta^{j'} \gamma^{j''} \), or the sum of all the products of three powers of the root. Now \( \Sigma \alpha^i \beta^{j'} \times S_{j''} \) will contain three sorts of terms, namely, products, such as \( \alpha^{i+j'+j''} \) and \( \alpha^{i+j'+j''} \), in which two roots only are combined, and the products of which the sum is required; therefore

\[ \Sigma \alpha^i \beta^{j'} \gamma^{j''} = \Sigma \alpha^i \beta^{j'} \times S_{j''} - \Sigma \alpha^{i+j'+j''} \beta^{j'} - \Sigma \alpha^{i+j'+j''} \beta^{j'} \]

but, according to the last case,

\[ \Sigma \alpha^{i+j'+j''} \beta^{j'} = S_{i+j'+j''} \times S_{j'} - S_{i+j'+j''} \] \[ \Sigma \alpha^{i+j'+j''} \beta^{j'} = S_{i+j'+j''} \times S_{j'} - S_{i+j'+j''}; \]

wherefore

\[ \Sigma \alpha^i \beta^{j'} \gamma^{j''} = S_i \times S_{j'} \times S_{j''} - S_{i+j'+j''} \times S_{j'} - S_{i+j'+j''} \times S_{j'} - S_{i+j'+j''} \times S_i + 2S_{i+j'+j''} \]

In like manner, when four different powers of the roots are multiplied together, we get

\[ \Sigma \alpha^i \beta^{j'} \gamma^{j''} \delta^{j'''} = \Sigma \alpha^i \beta^{j'} \gamma^{j''} \times S_{j'''} - \Sigma \alpha^{i+j'+j''} \beta^{j'} \gamma^{j'''} - \Sigma \alpha^{i+j'+j''} \beta^{j'} \gamma^{j'''} - \Sigma \alpha^{i+j'+j''} \beta^{j'} \gamma^{j'''} \]

and we have only to apply the preceding case, in order to obtain the expression of the quantity sought in terms of the sums of the like powers of the roots.

According to the procedure just explained, the case where any number of powers are multiplied together, is reduced to the simpler case where the powers multiplied are one less. There would be no great difficulty in deducing a general formula for the sum when the products contain any proposed number of different powers; but this would lead to calculations incompatible with the length of this article; and it may be doubted, whether the use of such a formula is preferable in any cases likely to occur in practice, to the application of the principles here laid down.

The theory of symmetrical functions is of the most extensive use in every branch of the doctrine of equations. Thus, if it be required to form an equation, the roots of which shall be any combinations of the roots of a given equation; it is manifest, that the coefficients of the equation sought will be symmetrical functions of the roots of the given equation; and hence they may be found, by calculating these functions in terms of the coefficients of the given equation.

The theory of symmetrical functions is also of use in approximating to the roots of numerical equations. Sir Isaac Newton seems to have had this application in view, in giving his rule for computing the sums of the like powers of the roots. He observes, that the powers of a great number increase in a much higher ratio than the same powers of less numbers; and hence, the \( 2r \)th power of the greatest root of an equation will approach nearer to the sum of the \( 2r \)th powers of all the roots, as \( r \) is greater. Wherefore, neglecting the distinction between positive and negative roots, if we calculate \( S_{2r} \) and then extract its \( 2r \)th root, we shall have an approximation to the root of the equation greatest in point of magnitude; and the approximation will be so much more accurate as \( r \) is greater.

But there is a more convenient way of approximating to the greatest and least roots of an equation, by means of symmetrical functions. For, since

\[ S_{r+1} = \alpha^{r+1} + \beta^{r+1} + \text{&c.} \] \[ S_r = \alpha^r + \beta^r + \text{&c.} \]

we have

\[ \frac{S_{r+1}}{S_r} = \frac{1 + \frac{\beta^{r+1}}{\alpha^{r+1}} + \text{&c.}}{1 + \frac{\beta^r}{\alpha^r} + \text{&c.}} \]

Now, \( \alpha \) being the greatest root, the fraction on the right-hand side will approach to unit when \( r \) is sufficiently large, in which case \( \frac{S_{r+1}}{S_r} \) will be nearly equal to \( \alpha \). Hence, if we compute a series of consecutive sums, viz. \( S_r, S_{r+1}, S_{r+2}, \) &c.; the values

\[ \frac{S_{r+1}}{S_r}, \frac{S_{r+2}}{S_{r+1}}, \frac{S_{r+3}}{S_{r+2}}, \text{&c.} \]

will approach nearer and nearer to the greatest root of the equation.

In like manner, if we take the sums of the negative powers of the roots, we shall have

\[ \frac{S_{-r}}{S_{-r-1}} = \alpha \cdot \frac{1 + \frac{\beta^r}{\alpha^r} + \text{&c.}}{1 + \frac{\beta^{r+1}}{\alpha^{r+1}} + \text{&c.}} \]

from which it appears that \( \frac{S_{-r}}{S_{-r-1}} \) will approximate so much more to \( \alpha \), the least root of the equation, as \( r \) is greater. Trinomial Divisors.

10. We proceed next to consider the trinomial divisors of a given polynomial; and, in order to avoid reference to other treatises, we shall begin with a short investigation of a preliminary point.

We have this identical expression,

\[ x^2 - y^2 = (x+y) \cdot (x-y); \]

consequently,

\[ (x^2 - y^2)^n = (x+y)^n \cdot (x-y)^n; \]

and, again,

\[ (x^2 - y^2)^n = \frac{1}{4} \left\{ (x+y)^n + (x-y)^n \right\}^2 \] \[ - \frac{1}{4} \left\{ (x+y)^n - (x-y)^n \right\}^2. \]

Now, using the letters H and G as the characteristics of the particular functions under consideration, let

\[ H_n(x,y^2) = \frac{1}{2} \left\{ (x+y)^n + (x-y)^n \right\} \] \[ G_n(x,y^2) = \frac{1}{2} \cdot \frac{(x+y)^n - (x-y)^n}{y}; \]

or, by expanding the binomial quantities in series,

\[ H_n(x,y^2) = x^n + n \cdot \frac{n-1}{2} x^{n-2} y^2 + \text{&c.} \] \[ G_n(x,y^2) = nx^{n-1} + n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} x^{n-3} y^2 + \text{&c.}; \]

then, by means of these notations, the preceding expression will be thus written, viz.

\[ (x^2 - y^2)^n = \left\{ H_n(x,y^2) \right\}^2 - y^2 \left\{ G_n(x,y^2) \right\}^2. \]

This equation is identical; that is, when the expressions on both sides of the sign of equality are expanded in series of terms containing the powers of \( y^2 \), they will consist of the same quantities with the same signs. It is evident, therefore, that the equation will still be identical, if we change \( y^2 \) into \( -y^2 \); for, by this change, the simple quantities of the developed expressions will not be affected; and no alteration will be produced, except in the signs of the odd powers of \( y^2 \), which will now be contrary to what they were before. We therefore have

\[ (x^2 + y^2)^n = \left\{ H_n(x,-y^2) \right\}^2 + y^2 \left\{ G_n(x,-y^2) \right\}^2, \]

in which equation it is to be observed that the functional expressions are not, as in the former instance, susceptible of an abridged algebraic notation, at least without introducing a new sign; but they can be exhibited in series, viz.

\[ H_n(x,-y^2) = x^n - n \cdot \frac{n-1}{2} x^{n-2} y^2 + \] \[ n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \cdot \frac{n-3}{4} x^{n-4} y^2 - \text{&c.} \] \[ G_n(x,-y^2) = nx^{n-1} - n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} x^{n-3} y^2 + \text{&c.} \]

Now put \( x = r \cos \varphi, y = r \sin \varphi, x^2 + y^2 = r^2 \); and let \( \varphi^{(n)} \) denote an arc, depending, in a certain manner, not yet discovered, upon the arc \( \varphi \) and the index \( n \); then, in consequence of the equation obtained above, we shall have

\[ r^n \cos \varphi^{(n)} = H_n(x,-y^2) \] \[ r^n \sin \varphi^{(n)} = y G_n(x,-y^2). \]

Again, multiply both sides of the same equation last referred to by \( x^2 + y^2 \); then

\[ (x^2 + y^2)^{n+1} = \left\{ x \cdot H_n(x,-y^2) - y^2 G_n(x,-y^2) \right\}^2 \] \[ + y^2 \left\{ H_n(x,-y^2) + x G_n(x,-y^2) \right\}^2 \]

but, since the equation alluded to is general for all the values of \( n \), we may write \( n+1 \) for \( n \); and thus we get

\[ (x^2 + y^2)^{n+1} = \left\{ H_{n+1}(x,-y^2) \right\}^2 + \] \[ y^2 \left\{ G_{n+1}(x,-y^2) \right\}^2; \]

therefore, by comparing the two values of \( (x^2 + y^2)^{n+1} \),

\[ H_{n+1}(x,-y^2) = x \cdot H_n(x,-y^2) - y^2 G(x,-y^2) \] \[ y G_{n+1}(x,-y^2) = y H_n(x,-y^2) + x G_n(x,-y^2): \]

and finally, by substituting the values of the functions in terms of the arcs, \( \varphi, \varphi^{(n)}, \varphi^{(n+1)} \), we shall obtain

\[ \cos \varphi^{n+1} = \cos \varphi \cos \varphi^{(n)} - \sin \varphi \sin \varphi^{(n)} = \cos (\varphi^{(n)} + \varphi) \] \[ \sin \varphi^{n+1} = \cos \varphi \sin \varphi^{(n)} + \sin \varphi \cos \varphi^{(n)} = \sin (\varphi^{(n)} + \varphi) \] \[ \varphi^{(n+1)} = \varphi^{(n)} + \varphi. \]

Now, if we make \( n \) successively equal to 1, 2, 3, &c. the results will be,

\[ \varphi^{(2)} = 2 \varphi \] \[ \varphi^{(3)} = 3 \varphi \] \[ \text{&c.} \]

and generally, \( \varphi^{(n)} = n \varphi \).

Thus it appears that

\[ r^n \cos n \varphi = H_n(x,-y^2) \] \[ r^{n-1} \times \frac{\sin n \varphi}{\sin \varphi} = G_n(x,-y^2); \]

or, if we take the expanded expressions of the functions,

\[ r^n \cos n \varphi = x^n - n \cdot \frac{n-1}{2} x^{n-2} y^2 + \] \[ n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} \cdot \frac{n-3}{4} x^{n-4} y^2 - \text{&c.} \] \[ r^{n-1} \times \frac{\sin n \varphi}{\sin \varphi} = nx^{n-1} - n \cdot \frac{n-1}{2} \cdot \frac{n-2}{3} x^{n-3} y^2 + \text{&c.} \]

in which formulae, \( x = r \cos \varphi, y = r \sin \varphi \).

The functions here designated by the letters H and G may be expressed by means of the imaginary sign; for we have

\[ H_n(x,-y^2) = \frac{(x+y \sqrt{-1})^n + (x-y \sqrt{-1})^n}{2} \] G_n(x, -y^2) = \frac{(x+y\sqrt{-1})^n-(x-y\sqrt{-1})^n}{2y\sqrt{-1}}:

And, in the case of \( r=1 \), the formulae obtained above are equivalent to the expressions known in analysis since the time of Dr Moivre, viz.

\[ \cos n\varphi = \frac{(\cos \varphi + \sin \varphi \sqrt{-1})^n + (\cos \varphi - \sin \varphi \sqrt{-1})^n}{2} \]

\[ \sin n\varphi = \frac{(\cos \varphi + \sin \varphi \sqrt{-1})^n - (\cos \varphi - \sin \varphi \sqrt{-1})^n}{2\sqrt{-1}} \]

But the mode of investigation we have followed is rigorous; and it has the advantage of leading to the true import of the imaginary sign, and of putting in a clear light its real effect in analytical operations. The real use of this sign may be shortly described by saying that it performs for even and odd functions the same office that the negative sign does for ordinary functions; in other words, when, by means of the ordinary operations of analysis, it has been proved that an even or odd function of an indeterminate quantity is equal to zero, it is by means of the impossible sign that the same equation is extended to the case when the square of the indeterminate quantity is negative. Every function of the indeterminate quantity \( x \) may be thus represented, viz.

\[ \varphi(x^2) \pm x\Psi(x^2); \]

and the substitution of \( x\sqrt{-1} \) in place of \( x \), has no other effect than to change the preceding expression into the one following, viz.

\[ \varphi(-x^2) \pm x\sqrt{-1} \cdot \Psi(-x^2): \]

and from this it is obvious, that the same operations which, in the one case, lead us to the equations \( \varphi(x^2)=0 \) and \( x\Psi(x^2)=0 \), will, in the other, necessarily conduct us to the equations \( \varphi(-x^2)=0 \) and \( x\Psi(-x^2)=0 \). It is to be observed, too, that the truth of the two latter equations is involved in that of the former. For the former equations cannot be generally true for all values of \( x^2 \), unless they are identical, or consist of equal quantities with opposite signs that mutually destroy one another; in which case the latter equations will also be identical. The sign of impossibility, as it has been called, is, therefore, one as truly significant as any other in analysis. It has, indeed, no consistent meaning when we consider it as only affecting \( x \), or the indeterminate quantity to which it is joined; but it becomes perfectly intelligible when we contemplate the real changes produced by it in the functions of even and odd dimensions, in which its conclusions are always ultimately expressed. When the true import and real effect of the imaginary sign are clearly apprehended, the truth of its conclusion is no longer doubtful or mysterious, but follows as a necessary consequence of a fundamental principle of analytical language. Proceeding on this principle we may even lay aside the imaginary character; and, in every particular case, with the assistance of a proper notation, arrive, by the ordinary operations, at the same conclusion to which it leads, as has been done in the preceding instance. It is to be observed further, that the imaginary arithmetic is not merely a short method of calculation convenient in practice, and that may be dispensed with; it is strictly a necessary branch of analysis, without which, or some equivalent mode of investigation, that science would be extremely imperfect. The equations \( \varphi(x^2)=0 \) and \( x\Psi(x^2)=0 \), are unchangeable by any operations with the signs commonly received, by the use of which alone it is impossible to deduce, in a direct manner, the related equations \( \varphi(-x^2)=0 \) and \( x\Psi(-x^2)=0 \); although the latter are equally true, of as frequent occurrence, and as extensive application, as the former. Without the impossible sign the operations of algebra would, therefore, be defective; since there are analytical truths that could not be investigated in a direct manner by means of the elementary signs usually admitted. It is to supply this defect that the Imaginary Arithmetic has been introduced, and has grown up to be an extensive branch of analysis; advancing at first by slow steps, because the true import of the character it employs, and the real effect of its operations, were neither clearly perceived nor fully understood. But, having premised what is conducive to our present purpose, we proceed to the investigation of the trinomial divisors of rational functions.

11. Every polynome of odd dimensions, having at least one binomial factor, it may, by dividing by that factor, be reduced to another polynome one degree lower. And hence, in this part of our subject, we may confine our attention to polynomes of even dimensions. We may also suppose that the even polynomes, under consideration have no double, triple, &c. factors of any kind; since, in case any such are present, they can be found separately and eliminated by division.

Suppose, then, that \( f(x) \) represents any polynome of even dimensions; let \( \xi-u \) be substituted in place of \( x \); and, by using the notation of the differential calculus, the given polynome will be transformed into

\[ f(\xi) + \frac{d f(\xi)}{d\xi} \cdot u + \frac{1}{2} \cdot \frac{d^2 f(\xi)}{d\xi^2} \cdot u^2 + \text{&c.} \]

Since \( f(x) \) is an even polynome, the equation \( \frac{d f(x)}{dx}=0 \) will be one of odd dimensions, having at least one root. Let \( \xi \) be the sole root of \( \frac{d f(x)}{dx}=0 \), when it has but one; and the greatest root, when it has several; then, because \( \frac{d f(\xi)}{d\xi}=0 \), the transformed function will become

\[ f(\xi) + \frac{1}{2} \cdot \frac{d^2 f(\xi)}{d\xi^2} \cdot u^2 + \frac{1}{6} \cdot \frac{d^3 f(\xi)}{d\xi^3} \cdot u^3 + \text{&c.} \]

It readily appears, from what was formerly proved (Sect. 8), that \( \xi \), the greatest root of \( \frac{d f(x)}{dx}=0 \), exceeds the greatest root of any of the equations,

\[ \frac{1}{2} \cdot \frac{d^2 f(x)}{dx^2} = 0, \quad \frac{1}{6} \cdot \frac{d^3 f(x)}{dx^3} = 0, \quad \text{&c.}; \text{ and, because,} \]

in any equation, the substitution of a value greater than the greatest root must give a positive result, all the quantities \( \frac{1}{2} \cdot \frac{d^2 f(e)}{d e^2}, \frac{1}{6} \cdot \frac{d^5 f(e)}{d e^5}, \) &c. will be positive. With regard to \( f(e) \) it may be either positive or negative, but not equal to zero; since this last case can happen only when the polynome has equal roots. The original polynome will, therefore, assume this form, viz.

\[ -y + A^{(2)} u^2 + A^{(3)} u^3 + A^{(4)} u^4 \ldots + A^{(2n-1)} u^{2n-1} + u^{2n}, \]

in which expression \( y, A^{(2)}, A^{(3)}, \) &c. represent any positive quantities.

The most interesting proposition in the branch of the subject under consideration, is to prove that every polynome of even dimensions has a quadratic divisor, either of the form \( (u+\alpha)^2-\tau^2 \), which admits two real binomial factors, or of the form \( (u-a)^2+\tau^2 \), which has two imaginary factors. By the preceding transformation this proposition is brought under two cases, according as \( y \) is affected with the sign minus or plus; the quadratic divisor being always of the form \( (u+\alpha)^2-\tau^2 \) in the first case; and always of the form \( (u-a)^2+\tau^2 \) in the other case; a distinction that agrees with what was before proved, Sect. 8.

Now the first of these cases is attended with no difficulty. For two values of \( u \), one negative and one positive, may be found that will satisfy the equation, Sect. 4.

\[ y = A^{(2)} u^2 + A^{(3)} u^3 + A^{(4)} u^4 \ldots + u^{2n} \]

Of these values, it is obvious that the negative one will be always greater than the positive one; and they may, therefore, be represented by \( -(\tau+\alpha) \) and \( \tau-\alpha \); wherefore, the polynome

\[ -y + A u^2 + A^{(3)} u^3 + A^{(4)} u^4 \ldots + u^{2n}, \]

will be divisible by each of the binomial factors,

\[ u + \tau + \alpha \\ u - \tau + \alpha; \]

and likewise by the quadratic factor,

\[ (u+\alpha)^2-\tau^2, \]

produced by their multiplication.

But the same mode of reasoning will not apply when \( y \) has the sign plus ; in which case the demonstration must be deduced from other principles.

12. If we put

\[ \varphi(u) = A^{(2)} u^2 + A^{(3)} u^3 + A^{(4)} u^4 \ldots + u^{2n}, \]

the transformed polynome, supposing \( y \) to have the sign plus, will become

\[ y + \varphi(u). \]

Let \( (u-\alpha)^2+\tau^2 \) be a quadratic divisor of this polynome, and put \( u-\alpha=z \), or \( u=\alpha+z \); then, by substituting \( \alpha+z \) for \( u \), and writing all the terms of the transformed function \( \varphi(\alpha+z) \) in two lines, one containing all the even, and the other all the odd, powers of \( z \); the polynome \( y+\varphi(u) \) will be equal to.

\[ y + \varphi(a) + \frac{1}{2} \cdot \frac{d^2 \varphi(a)}{da^2} \cdot z^2 + \frac{1}{24} \cdot \frac{d^4 \varphi(a)}{da^4} \cdot z^4 + \text{&c.} \\ + z \left\{ \frac{d \varphi(a)}{da} + \frac{1}{6} \cdot \frac{d^3 \varphi(a)}{da^3} \cdot z^2 + \text{&c.} \right\}. \]

By the same substitution of \( z \) for \( u-\alpha \), the divisor \( (u-\alpha)^2+\tau^2 \) is changed into the binomial quantity \( z^2+\tau^2 \); which will be a divisor of each of the preceding lines, if \( -\tau^2 \), when it is substituted for \( z^2 \), render each of them equal to zero, Sect. 3. Hence we obtain the two following equations, viz.

\[ o = y + \varphi(a) - \frac{1}{2} \cdot \frac{d^2 \varphi(a)}{da^2} \cdot r^2 + \frac{1}{24} \cdot \frac{d^4 \varphi(a)}{da^4} \cdot r^4 + \text{&c.} \\ \frac{d \varphi(a)}{da} - \frac{1}{6} \cdot \frac{d^3 \varphi(a)}{da^3} \cdot r^2 + \text{&c.} \] (C)

If two numbers, \( a \) and \( \tau^2 \), can be found that will satisfy these equations, it is evident that \( z^2+\tau^2 \) will be a divisor of each of the two lines that compose the transformed function \( y+\varphi(a+z) \); consequently, it will be a divisor of the sum of both lines, or of the function itself; that is, \( (u-\alpha)^2+\tau^2 \) will be a divisor of the proposed polynome \( y+\varphi(u) \). We are now to prove that two such numbers may be found.

Substitute \( \lambda^2 a^2-s \) for \( r^2 \) in the equations (C), \( \lambda \) being a quantity to be afterwards determined; and, in order to shorten expressions, put

\[ M = \varphi(a) - \frac{1}{2} \cdot \frac{d^2 \varphi(a)}{da^2} (\lambda^2 a^2-s) + \frac{1}{24} \cdot \frac{d^4 \varphi(a)}{da^4} + (\lambda^2 a^2-s)^2 - \text{&c.} \\ N = \frac{d \varphi(a)}{da} - \frac{1}{6} \cdot \frac{d^3 \varphi(a)}{da^3} (\lambda^2 a^2-s) + \text{&c.} \]

And the two equations (C) will be thus written, viz.

\[ y + M = o \\ N = o. \]

In these equations \( a \) and \( s \) are always supposed to represent positive numbers, in which case the equation \( N=o \) cannot take place when \( s \) is greater than \( \lambda^2 a^2 \); for then all the terms of \( N \) would be positive.

Considering \( N \) as a function of \( a \), the part of it that does not contain \( a \) is evidently

\[ A^{(3)} s + A^{(5)} s^2 + A^{(7)} s^3 + \text{&c.} \]

which is always positive. The highest power of \( a \) contained in the same function is \( a^{2n-1} \); and we shall obtain all the terms of \( N \) that contain this power by putting \( a^{2n} \) for \( \varphi(a) \) in the expression,

\[ \frac{d \varphi(a)}{da} - \frac{1}{6} \cdot \frac{d^3 \varphi(a)}{da^3} \lambda^2 a + \frac{1}{120} \cdot \frac{d^5 \varphi(a)}{da^5} \lambda^4 - \text{&c.}; \]

which terms are therefore as follows, viz.

\[ a^{2n-1} \left\{ \begin{array}{l} 2n-2n. \frac{2n-1.2n-2}{2.3} \lambda^2 + 2n. \\ \frac{2n-1.2n-2.2n-3.2n-4}{2.3.4.5} \lambda^4 + \text{&c.} \end{array} \right. \]

Now, in the expression obtained in Sect. 10, viz.

\[ \frac{x^{2n-1}}{\sin \varphi} \times \frac{\sin 2n \varphi}{\sin \varphi} = 2nx^{2n-1} - 2n. \frac{2n-1.2n-2}{2.3} x^{2n-3} y^2 + \text{&c.} \]

if we put \( \lambda^2 = \frac{y^2}{x^2} \tan^2 \varphi \), and divide both sides by \( x^{2n-1} = x^{2n-1} \cos^{2n-1} \varphi \); we shall obtain: Equations.

from which formula it follows, that the polynome on the right-hand side of the sign of equality will be equal to nothing, where \( \varphi = \pm \frac{m}{n} \times 90^\circ \), \( m \) being any integer number less than \( n \), zero not included. Wherefore the first, third, &c. roots of the polynome will be expressed by the formula

\[ \lambda^2 = \tan \frac{2k+1}{n} \cdot 90^\circ, \]

\( 2k+1 \) representing any odd number less than \( n \); and the second, fourth, &c. roots by the formula

\[ \lambda^2 = \tan \frac{2k+2}{n} \cdot 90^\circ, \]

\( 2k+2 \) being any even number less than \( n \). And it is evident that the polynome will be negative for every value of \( \lambda^2 \) that lies between any odd root and the next even root, that is, for every value between these limits, viz.

\[ \lambda^2 > \tan \frac{2k+1}{n} \cdot 90^\circ \\ \lambda^2 < \tan \frac{2k+2}{n} \cdot 90^\circ. \]

Thus, an indefinite number of values of \( \lambda^2 \) may be found that will make the polynome negative.

Having assumed such a value of \( \lambda^2 \), let any positive number whatever be substituted for \( s \), and N will be converted into a rational function of \( \alpha \); the greatest power of \( \alpha \), or \( \alpha^{2n-1} \), being odd, and having a negative coefficient; and the term which does not contain \( \alpha \) being positive. Therefore, at least, one positive value of \( \alpha \) may be found that will satisfy the equation \( N=0 \); and, as has already been observed, this value of \( \alpha \) will be such as to make \( \lambda^2 \alpha^2 \) —s a positive quantity. It is possible indeed that, in the equation \( N=0 \), there may be several values of \( \alpha \) for every assumed value of \( s \); but we here confine our attention to the least positive value, which is distinguished by this circumstance, that it vanishes with the absolute term of the equation, or with \( s \); whereas, when \( s \) is equal to zero, all the other roots of the equation \( N=0 \) have finite values depending upon the given coefficients.

Now, if we suppose \( s \) to increase from zero to infinity, and assume two values, \( s \) and \( s+\delta s \), very near one another, according to what has been proved, we shall have the corresponding values, \( \alpha \) and \( \alpha+\delta \alpha \), such, that the equation \( N=0 \) will be satisfied by substituting both \( s \) and \( \alpha \), and likewise \( s+\delta s \) and \( \alpha+\delta \alpha \). Hence, because \( N=0 \), and \( \delta N=0 \), we get

\[ \frac{dN}{da} \cdot \delta \alpha + \frac{dN}{ds} \cdot \delta s = 0 \]

and, \( \delta \alpha = - \left( \frac{dN}{da} \right) \times \frac{dN}{ds} \).

Again, if we substitute first \( s \) and \( \alpha \), and then \( s+\delta s \) and \( \alpha+\delta \alpha \), in the function M, we shall get

\[ \delta M = \frac{dM}{da} \cdot \delta \alpha + \frac{dM}{ds} \cdot \delta s. \]

But, by comparing the functions M and N, the following properties will readily be discovered, viz.

\[ \frac{dM}{da} + 2\lambda^2 a \cdot \frac{dM}{ds} = N - 2 \frac{dN}{ds} (\lambda^2 a^2 - s) \] \[ \frac{dM}{ds} = \frac{1}{2} \cdot \frac{dN}{da} + \lambda^2 a \cdot \frac{dN}{ds} : \]

whence,

\[ \frac{dM}{da} = N - 2 \frac{dN}{ds} (\lambda^2 a^2 - s) - \frac{dN}{da} \lambda^2 a - 2\lambda^4 a^2 \frac{dN}{ds}. \]

Consequently,

\[ \delta M = \left\{ N - 2 \frac{dN}{ds} (\lambda^2 a^2 - s) - \frac{dN}{da} \lambda^2 a - 2\lambda^4 a^2 \frac{dN}{ds} \right\} + \delta \alpha + \left\{ \frac{1}{2} \cdot \frac{dN}{da} + \lambda^2 a \cdot \frac{dN}{ds} \right\} \cdot \delta s: \]

and, if we observe that \( N=0 \), and substitute the value of \( \delta \alpha \) found above, we shall get

\[ \delta M = - \frac{\delta s}{\left( \frac{dN}{da} \right)} \cdot \left\{ 2 \left( \frac{dN}{ds} \right)^2 (\lambda^2 a^2 - s) + \frac{1}{2} \left( \frac{dN}{da} + 2\lambda^2 a \frac{dN}{ds} \right)^2 \right\} \]

in which expression all the quantities are essentially positive, except \( \frac{dN}{da} \), which is always negative, as may be thus proved.

The quantity \( s \) remaining invariable, if we make \( \alpha=0 \), the function N will be positive; for it is equal to

\[ A^{(3)} s + A^{(5)} s^3 + A^{(7)} s^5 + \ldots \]

and the same function will continue positive, while \( \alpha \) increases from zero to the least root of the equation \( N=0 \). At this limit, N is first equal to zero, and then becomes negative; it must, therefore, be decreasing, and consequently \( \frac{dN}{da} \) is negative. It may indeed happen, that, for particular values of \( s \), the coefficients of N may be such, that N and \( \frac{dN}{da} \) shall be both equal to zero at the same time; but, in such cases, it will readily appear, that \( \frac{dN}{ds} \) and \( \delta M \) will likewise be equal to zero. Wherefore \( \delta M \) will be negative; at least, if it become equal to zero for any particular values of \( s \) and \( \alpha \), it cannot become positive. It follows, therefore, that the function M itself will be invariably negative, while \( s \) and \( \alpha \) increase together from zero to be infinitely great.

Now assume a series of values of \( s \) increasing from zero without limit, viz.

\[ 0, s^{(1)}, s^{(2)}, s^{(3)}, \ldots, s^{(x)}, s^{(x+1)}, \ldots \]

and having substituted these in the function N, find, by means of the equation \( N=0 \), the corresponding values of \( \alpha \), viz. then, by substituting these values in M, we shall obtain a series of results all negative, and increasing from zero without limit, viz.

\( o, -M^{(1)}, -M^{(2)}, -M^{(3)} \ldots -M^{(x)}, -M^{(x+1)} \ldots \)

and whatever be the magnitude of the positive quantity \( y \), it must be contained between two consecutive terms of this last series, viz. between \( M^{(x)} \) and \( M^{(x+1)} \). But as the values of \( s \) may be assumed as near one another as we please, it follows that \( M^{(x)} \) and \( M^{(x+1)} \) may be made to approach to one another and to \( y \), within any required degree of accuracy. Thus, two values of \( s \) and \( a \) may be found that will satisfy both the equations,

\[ \frac{y + M}{N} = o \\ N = o : \]

and having found these values, we shall obtain the quadratic divisor of the proposed polynome \( y + \varphi(u) \), viz. \( (u - a)^2 + r^2 \); or

\[ (u - a)^2 + \lambda^2 a^2 - s. \]

In the preceding demonstration, it is supposed, that M increases without limit, as \( s \) becomes indefinitely great; which may be thus proved: The values of M and N will coincide nearly with the terms containing the highest powers of \( s \) and \( a \), when these quantities are very great; and ultimately the functions may be considered as equal to those terms alone. In such circumstances, therefore, the values of the functions will be found by writing \( a^{2n} \) for \( \varphi(a) \); whence we get

\[ M = a^{2n} - 2n \cdot \frac{2n-1}{2} \cdot a^{2n-2} (\lambda^2 a^2 - s) + &c. \] \[ N = 2na^{2n-1} - 2n \cdot \frac{2n-1}{2} \cdot \frac{2n-2}{3} (\lambda^2 a^2 - s) + &c.; \]

and if we put \( \lambda^2 a^2 - s = t^2 a^2 \), or \( a^2 = \frac{s}{\lambda^2 - t^2} \); then,

\[ M = a^{2n} \times \left\{ 1 - 2n \cdot \frac{2n-1}{2} \cdot t^2 + &c. \right\} \] \[ N = a^{2n-1} \times \left\{ 2n - 2n \cdot \frac{2n-1}{2} \cdot \frac{2n-2}{3} \cdot t^2 + &c. \right\}. \]

Now, \( s \) remaining invariable, \( a \) will increase as \( t^2 \) increases; and the least value of \( a \) that will satisfy the equation \( N = o \), corresponds to the least value of \( t^2 \) that will make the polynome in the expression of N equal to zero; which value, according to what was before shown, is

\[ t^2 = \tan^2 \frac{1}{n} \times 90^\circ. \]

But, if we put \( t = \tan \varphi \), we shall get

\[ M = a^{2n} \times \frac{\cos 2n \varphi}{\cos 2n \varphi}; \]

or, because \( \varphi = \frac{1}{n} \times 90^\circ \); \( \cos \varphi = \frac{1}{\sqrt{1 + t^2}} \), and \( a^2 = \frac{s}{\lambda^2 - t^2} \);

\[ M = -s^n \times \left( \frac{1 + t^2}{\lambda^2 - t^2} \right)^n; \]

which proves the point assumed in the demonstration.

By a similar mode of reasoning, we may likewise prove the former case of the proposition, when \( y \) is negative. In this case, the quadratic divisor is \( (u - a)^2 - r^2 \); and if we proceed as before, or, which is the same thing, if we change the signs of \( y \) and \( r^2 \) in the equations (C) already obtained, and put

\[ M = \varphi(a) + \frac{1}{2} \cdot \frac{d^2 \varphi(a)}{da^2} \cdot r^2 + &c. \] \[ N = \frac{d \varphi(a)}{da} + \frac{1}{6} \cdot \frac{d^3 \varphi(a)}{da^3} r^2 + &c.; \]

we shall get

\[ -y + M = o \\ N = o. \]

Now, by pursuing the steps of the foregoing analysis, we may prove, first, that, for every assumed value of \( r^2 \), a negative value of \( a \) may be found, which will satisfy the equation \( N = o \); and, secondly, that, when the values which satisfy the equation \( N = o \) are substituted in the function M, the results will be invariably positive: whence it follows that a positive value of \( r^2 \), and a negative value of \( a \), may be found that will satisfy both the equations, whatever be the magnitude of \( y \). The analogy between the two cases is thus placed in a strong light; and a little reflection will even bring us to this conclusion, that in reality the one case is a necessary consequence of the other. For since \( a \) and \( r^2 \) depend only upon \( y \), and the given coefficients of the polynome, they will be functions of \( y \); wherefore, in the equations of the first case, viz.

\[ -y + M = o \\ N = o, \]

\( a \) being negative, and \( r^2 \) positive, we may suppose \( -a = y \varphi(y) \) and \( r^2 = y \Psi(y) \), these values being such as to render each of the equations identical: and then the quadratic divisor \( (u - a)^2 - r^2 \) will become

\[ \left\{ u + y \varphi(y) \right\}^2 - y \Psi(y). \]

But, because the foregoing equations become identical by the substitution of the values mentioned, it is a necessary consequence that the equations of the second case, viz.

\[ y + M = o \\ N = o, \]

in which the signs of \( y \), \( a \), and \( r^2 \), are contrary to what they were in the former equations, will likewise be identical, when \( -a = -y \varphi(-y) \) and \( r^2 = -y \Psi(-y) \); and the quadratic divisor, \( (u - a)^2 - r^2 \), will now become

\[ \left\{ u - y \varphi(-y) \right\}^2 + y \Psi(-y). \]

Thus when the quadratic divisor of the first case is expressed in terms of \( y \), we have only to change the sign of that quantity, in order to have the quadratic divisor of the second case. It is not difficult to perceive, that what has now been proved is nothing more than another application of the principle employed in Sect. 10; a principle which is the real foundation Equations of the imaginary arithmetic, with the processes of which the preceding investigations are intimately connected. None but real quantities have occurred in the analysis we have pursued, because we have sought to investigate \( \tau^2 \) which is always rational; whereas, if we had proposed to find \( \tau \), we should inevitably have been led to the real quantity \( \sqrt{y} \) in the one case, and to the impossible quantity \( \sqrt{-y} \), in the other. These few observations are made for the purpose of throwing light upon a part of analysis, which is certainly obscure in its principles, although there is no question that it is a useful and even a necessary branch of the art of calculation. A fuller elucidation of the subject would be unsuitable to this place; but enough has been said to show that we must seek in the principles of analysis itself for the explanation of the operations it employs; and we may, with great probability, conclude, that no satisfactory account of the imaginary calculus will ever be obtained by having recourse to fanciful geometrical constructions, or to the analogy between the circle and the hyperbola, or to the metaphysical proposition, that all processes with general symbols, whether significant or not, are equally entitled to be considered as demonstrative.

13. Having now proved, in a rigorous manner, that every polynome of even dimensions has at least one quadratic divisor of the one kind or the other, it follows, that it may be reduced by division to another polynome two degrees lower: in like manner, this last polynome will admit of being lowered two degrees more; and by repeating the same process, the first polynome will at length be completely exhausted by quadratic divisors. If, therefore, we recollect, that every polynome of odd dimensions has one binomial divisor, we shall arrive at this general conclusion, "That every rational polynome can be completely exhausted by binomial and trinomial divisors; and, consequently, that it is equal to the product of a certain number of factors of the two first degrees."

It appears also that the binomial factors of any polynome are such only as arise from the resolution of the quadratic divisors; and they are, therefore, either real or imaginary. And thus we finally obtain the following proposition, which was assumed by Harriot, and is the foundation of the received theory of equations, namely, "Every rational polynome has as many binomial factors, and as many roots, real and imaginary, as it has dimensions."

The necessity of confirming, by a general demonstration, the assumed theory of the impossible roots of equations, was early felt; and, accordingly, this point has engaged the attention of all the great mathematicians to whom analysis is indebted for the progress it has made in the course of the last and the present centuries. An account of their several researches would greatly exceed the limits of this article; but the reader will find all the information he can wish for in two long notes (9 and 10) of the Traité des Equations Numeriques, by La Grange, in which the author, with his usual elegance, has explained and commented upon the various modes of investigation that have been proposed. It will be sufficient to observe here, that all the demonstrations that have appeared are either calculations with impossible quantities, or they proceed upon the assumption, that every equation has as many roots as dimensions, and thus involve the very thing to be proved.

14. The general cases in which mathematicians have been successful in resolving rational functions into their trinomial factors, are confined to the theorem of Cotes, and to a more general proposition of a similar kind, for which we are indebted to De Moivre. These instances are of great importance in analysis, and we shall therefore subjoin an investigation of them, because they are deduced in a very direct manner from the method we have followed.

Suppose, as before, that \( f(x) \), or \( x^n + A^{(1)} x^{n-1} + A^{(2)} x^{n-2} \ldots + A^{(n-1)} x + A^{(n)} \), is a rational polynome of \( n \) dimensions, and \( (x-\alpha)^2 + \tau^2 \) one of its quadratic divisors; put \( z = x - \alpha \), substitute \( \alpha + z \) for \( x \), and write the transformed function in two lines, one containing all the even, and the other all the odd powers of \( z \); then the polynome will be equal to

\[ f(a) + \frac{1}{2} \cdot \frac{d^2 f(a)}{da^2} z^2 + \frac{1}{24} \cdot \frac{d^4 f(a)}{da^4} z^4 + \text{&c.} \]

\[ + z \times \left\{ \frac{df(a)}{da} + \frac{1}{6} \cdot \frac{d^3 f(a)}{da^3} z^2 + \frac{1}{120} \cdot \frac{d^5 f(a)}{da^5} z^4 + \text{&c.} \right\} \]

By the same substitution of \( z \) for \( x - \alpha \), the divisor \( (x-\alpha)^2 + \tau^2 \) will become \( z^2 + \tau^2 \); and, as before, the conditions that \( z^2 + \tau^2 \) shall divide each of the foregoing lines, will be expressed by the following equations, viz.

\[ o = f(a) - \frac{1}{2} \cdot \frac{d^2 f(a)}{da^2} \tau^2 + \frac{1}{24} \cdot \frac{d^4 f(a)}{da^4} \tau^4 + \text{&c.} \tag{D} \]

\[ o = \frac{df(a)}{da} - \frac{1}{6} \cdot \frac{d^3 f(a)}{da^3} \tau^2 + \frac{1}{120} \cdot \frac{d^5 f(a)}{da^5} \tau^4 + \text{&c.} \]

In these formulae substitute the expanded values of \( f(a), \frac{df(a)}{da}, \text{&c.} \), and class together all the homogeneous terms of the same order, that is, all the terms in which the exponents of \( a \) and \( \tau \) amount to the same sum, then we shall have

\[ o = a_n - \frac{n-1}{2} a^{n-2} \tau^2 + \text{&c.} \]

\[ + A^{(1)} \left\{ a^{n-1} - n-1, \frac{n-2}{2} a^{n-3} \tau^2 + \text{&c.} \right. \]

\[ + A^{(2)} \left\{ a^{n-2} - n-2, \frac{n-3}{2} a^{n-4} \tau^2 + \text{&c.} \right. \]

\[ \text{&c.} \]

\[ o = n a^{n-1} - \frac{n-1}{2} \cdot \frac{n-2}{3} a^{n-3} \tau^2 + \text{&c.} \]

\[ + A^{(1)} \left\{ (n-1) a^{n-2} - n-1, \frac{n-2}{2} \cdot \frac{n-3}{3} a^{n-4} \tau^2 + \text{&c.} \right. \]

\[ + A^{(2)} \left\{ (n-2) a^{n-3} - n-2, \frac{n-3}{2} \cdot \frac{n-4}{3} a^{n-5} \tau^2 + \text{&c.} \right. \] Equations. Now, put \( \alpha = r \cos \varphi, \tau = r \sin \varphi \); and, by what was proved in Sect. 10, the two foregoing equations will become

\[ r^n \cos n \varphi + A^{(1)} r^{n-1} \cos (n-1) \varphi \ldots \ldots + A^{(n-1)} r \cos \varphi + A^{(n)} = 0 \tag{E} \]

\[ \frac{1}{\sin \varphi} \left\{ r^{n-1} \sin n \varphi + A^{(1)} r^{n-2} \sin (n-1) \varphi \ldots \ldots + A^{(n-1)} \sin \varphi \right\} = 0: \]

And the quadratic divisor \((x-\alpha)^2 + \tau^2\) will be changed into

\[ x^2 - 2r \cos \varphi x + r^2. \]

When \(\sin \varphi = 0\), and \(\varphi = 0\) or \(180^\circ\), the preceding equations coincide with these, viz.

\[ r^n \pm A^{(1)} r^{n-1} + A^{(2)} r^{n-2} \pm \text{c.c.} = 0 \] \[ nr^{n-1} \pm (n-1)A^{(1)} r^{n-2} + (n-2)A^{(2)} r^{n-3} \pm \text{c.c.} = 0, \]

which express the condition that the given polynome has two or more factors equal to \(x \mp r\); at which limits a quadratic divisor changes from being of the form \((x-a)^2 + r^2\) to be of the form \((x-a)^2 + r^2\), or the contrary. Thus we learn that, in the equations (E), \(\sin \varphi\) must always have a finite value, and then the denominator of the second equation may be neglected.

Let the preceding investigation be applied to find the quadratic factors of \(x^n - a^n\). In this case the two equations (E) will become

\[ r^n \cos n \varphi - a^n = 0 \] \[ r^{n-1} \times \frac{\sin n \varphi}{\sin \varphi} = 0: \]

Whence

\[ r = a \\ \cos n \varphi = 1 \\ \frac{\sin n \varphi}{\sin \varphi} = 0. \]

Now, excluding the cases when \(\varphi = 0\) and \(\varphi = 180^\circ\), the last equation will be satisfied when \(\varphi = \frac{2k+1}{n} \times 180^\circ\), or \(\varphi = \frac{2k}{n} \times 180^\circ\), the numerators of the fractions representing all the odd and even numbers less than the common denominator; but the second equation will be satisfied only when \(\varphi = \frac{2k}{n} \times 180^\circ\): wherefore all the quadratic factors of the function \(x^n - a^n\) will be comprehended in the formula

\[ x^2 - 2ax \times \cos \frac{2k}{n} \times 180^\circ + a^2. \]

When \(n\) is even number, the quadratic factors will amount to \(\frac{n-2}{2}\); and if to them we add the simple factors \(x + a\) and \(x - a\), we shall have the complete resolution of the function. When \(n\) is odd, the number of quadratic factors is \(\frac{n-1}{2}\), to which must be added the binomial factor \(x - a\).

By proceeding in a similar manner in the case of the function \(x^n + a^n\), we shall have the equations

\[ r = a \\ \cos n \varphi = -1 \\ \frac{\sin n \varphi}{\sin \varphi} = 0. \]

Excluding the cases when \(\varphi = 0\) and \(\varphi = 180^\circ\), the second and third equations will be both satisfied, when \(\varphi = \frac{2k+1}{n} \times 180^\circ\), the numerator of the fraction representing any odd number less than \(n\). Wherefore all the quadratic factors will be comprehended in the formula

\[ x^2 - 2ax \times \cos \frac{2k+1}{n} \times 180^\circ + a^2. \]

When \(n\) is even, the number of quadratic factors is \(\frac{n}{2}\) and they exhibit the complete resolution of the function. When \(n\) is odd, the number of quadratic factors is \(\frac{n-1}{2}\), to which the binomial factor \(x + a\) must be added.

Let us next take the more general function

\[ x^{2n} - 2\beta x^n a^n + a^{2n}. \]

And, in the first place, when \(\beta\) is greater than unit, the function is equal to

\[ \left\{ x^n - a^n (\beta + \sqrt{\beta^2 - 1}) \right\} \times \left\{ x^n - a^n (\beta - \sqrt{\beta^2 - 1}) \right\}; \]

and the quadratic factors may be found by the cases already considered.

When \(\beta\) is less than unit, let \(\beta = \cos \theta\), and the function to be resolved will be

\[ x^{2n} - 2a^n x^n \cos \theta + a^{2n}. \]

By means of the equations (E) we get

\[ r^{2n} \cos 2n \varphi - 2a^n r^n \cos \theta \cos n \varphi + a^{2n} = 0 \] \[ r^{2n-1} \times \frac{\sin 2n \varphi}{\sin \varphi} - 2a^n r^{n-1} \times \frac{\sin n \varphi}{\sin \varphi} \times \cos \theta = 0: \]

And hence

\[ r = a \\ \cos 2n \varphi - 2 \cos \theta \cos n \varphi + 1 = 0 \\ \frac{\sin 2n \varphi}{\sin \varphi} - 2 \frac{\sin n \varphi}{\sin \varphi} \times \cos \theta = 0. \]

But, \(\cos 2n \varphi + 1 = 2 \cos^2 n \varphi\); and \(\sin 2n \varphi = 2 \cos n \varphi \times \sin n \varphi\); wherefore the two last equations will become

\[ 2 \cos n \varphi (\cos n \varphi - \cos \theta) = 0 \\ 2 \frac{\sin n \varphi}{\sin \varphi} (\cos n \varphi - \cos \theta) = 0; \] and these, supposing cos \( \theta \) different from unit, can be satisfied only by making \( \cos n\varphi = \cos \theta = 0 \), or \[ \cos n\varphi = \cos \theta. \] Now, \( \cos \theta = \cos(m \times 360^\circ + \theta) \), \( m \) being any integer number whatever, zero included; and hence \( \varphi = \frac{m \times 360^\circ + \theta}{n} \), which formula comprehends all the values of \( \theta \) that will satisfy the above equations. Wherefore all the factors sought will be contained in this general expression, viz.

\[ x^2 - 2ax \cos \frac{m \times 360^\circ + \theta}{n} + a^2; \]

in which, if for \( m \) we substitute all the integer numbers less than \( n \), zero included, we shall obtain the \( n \) quadratic factors of the proposed function.

15. The quadratic divisors \( (x-a)^2 - s^2 \) and \( (x-a)^2 + r^2 \), have hitherto been considered separately; but they may be both represented by \( (x-a)^2 - s \), which will coincide with the one or the other according as \( s \) is positive or negative. And, if we now proceed as before, we shall get the following equations which express the conditions necessary, in order that the polynome \( f(x) \) of any proposed dimensions, as \( n \), shall be divisible by \( (x-a)^2 - s \), viz.

\[ o = f(a) + \frac{1}{2} \cdot \frac{d^2f(a)}{da^2} s + \frac{1}{24} \cdot \frac{d^4f(a)}{da^4} s^2 + &c. \]

\[ o = \frac{df(a)}{da} + \frac{1}{6} \cdot \frac{d^3f(a)}{da^3} s + \frac{1}{120} \cdot \frac{d^5f(a)}{da^5} s^2 + &c. \]

By eliminating \( s \) we shall obtain an equation, viz.

\[ A = o, \]

in which \( a \) is the unknown quantity. As the process of elimination is independent of the particular values of the coefficients of \( f(x) \), the degree of the resulting equation will be the same when the polynome \( f(x) \) has as many real roots as dimensions, and when the case is otherwise. But when \( f(x) \) is equal to the product of \( n \) real binomial factors, the multiplication of every two of them will form a quadratic factor. The number of such factors will, therefore, be equal to \( n \times \frac{n-1}{2} \), which expresses all the combinations made with \( n \) things taken two and two. Consequently, there will be just so many different values of \( a \) that will satisfy the equation \( A = o \), which will, therefore, have its exponent equal to \( n \times \frac{n-1}{2} \).

It thus appears that the equation \( A = o \) rises in its dimensions very rapidly above the given polynome, on which account little advantage is derived from this procedure.

Again, by eliminating \( a \) from the same two equations we shall obtain one, viz.

\[ S = o, \]

in which \( s \) is the unknown quantity. This equation, which has already been alluded to (Sect. 8), rises to the same dimensions with the former equation \( A = o \); but it is possessed of some useful properties, derived chiefly from the consideration, that every positive root gives a quadratic factor of the form

\[ (x-a)^2 - r^2 \]

in the polynome \( f(x) \), and every negative root, a quadratic factor of the form \( (x-a)^2 + r^2 \) in the same polynome.

The quadruple of \( s \) is equal to the square of the difference of the two binomial factors of \( (x-a)^2 - s \): whence it follows that the quadruples of the several roots of the equation \( S = o \) are equal to the squares of the differences of the roots of \( f(x) = o \). If, therefore, we put \( x', x'', x''', \ldots \) &c. for the roots of \( f(x) = o \), the roots of \( S = o \) will be

\[ \frac{1}{4}(x' - x'')^2, \frac{1}{4}(x' - x''')^2, \frac{1}{4}(x'' - x''')^2, \ldots \]

and from this it is manifest, that the coefficients of the same equation will be known symmetrical functions of the quantities \( x', x'', x''' \), &c. or of the roots of \( f(x) = o \). The rules formerly explained may, therefore, be employed for calculating the coefficients of \( S = o \); and this method of forming the equation is not only more convenient than the process of eliminating; but it likewise has the advantage of enabling us to find any one coefficient separately without computing the rest. Thus, if we put

\[ K^{(n)} = (x' - x'')^2, (x' - x''')^2, (x'' - x''')^2, \ldots \]

and expand this product, and in place of the symmetrical functions of which it is composed, substitute their values in terms of the given coefficients of \( f(x) = o \), we shall obtain the value of \( K^{(n)} \); and the last term of the equation \( S = o \) will be equal to

\[ \pm \frac{K^{(n)}}{2^{n(n-1)}} \]

the upper sign taking place when \( n \times \frac{n-1}{2} \), the dimensions of the equation \( S = o \), is even, and the lower sign when the same number is odd.

If we suppose the given equation \( f(x) = o \) to be possessed of as many real roots as dimensions, or to have \( n \) real binomial factors, the product of every two of these will be a quadratic factor \( (x-a)^2 - s \), in which \( s \) is positive; wherefore, the roots of \( S = o \) will be all real and all positive. On the other hand, when the given equation \( f(x) = o \) has not as many real roots as dimensions, it will be divisible by one or more quadratic factors not resolvable into real binomial factors, and in which \( s \) is negative; consequently, the equation \( S = o \) will have one or more negative roots. It is, therefore, a property of the auxiliary equation \( S = o \), that when the roots are all real, they are all positive; and when they are not all real, some of them are negative. Now the rule of Descartes will enable us to find whether the roots are all positive or not; and by this means we shall discover whether the roots of the given equation \( f(x) = o \) are all real or not. From what has been said we may lay down this rule: "The proposed equation \( f(x) = o \) will have all its roots real, when the auxiliary equation \( S = o \) has as many variations from one sign to another as it has dimensions, or when its terms are alternately positive and negative; otherwise the proposed equation will have one or more quadratic factors of the form \( (x-a)^2 + r^2 \), but the number of such factors cannot exceed the continuations of the same sign in the auxiliary equation."

Again, in the equation \( S = 0 \), the polynome \( S \) is equal to a certain number of binomial factors of the forms \( x - a \) and \( x + a \), multiplied into a supplementary polynome of even dimensions, which, not being capable of having a negative value, will have its last term positive (Sect. 5). It is manifest, therefore, that the last term of \( S = 0 \) will be positive or negative, according as the number of factors of the form \( x - a \) is even or odd, that is, according as the equation has an even or odd number of real and positive roots. But every two real roots in the equation \( f(x) = 0 \) give one real and positive root in the subsidiary equation \( S = 0 \); wherefore, if \( m \) denote the number of real roots in the former equation, the number of real and positive roots in the latter will be equal to

\[ m \times \frac{m-1}{2}; \]

and the last term of the subsidiary equation will be positive or negative, according as

\[ m \times \frac{m-1}{2} \]

is an even or an odd number.

In a cubic equation \( x^3 + px + q = 0 \), \( m \) is either one or three. In the first case, the equation \( S = 0 \) will have no positive roots, and the last term will be positive; in the second case, it will have three real and positive roots, and the last term will be negative. Now, the dimensions of \( S = 0 \) being odd, the function \( K^{(3)} \) will be negative in the first case, and positive in the second. Wherefore the given cubic equation will have one real root, or three, according as the function \( K^{(3)} \), that is,

\[ (x' - x'')^2, (x' - x''')^2, (x'' - x''')^2, \]

or \(-4p^3 - 27q^2\), is negative or positive.

In a biquadratic equation \( x^4 + px^2 + qx + r = 0 \), \( m \) is equal to zero, or two, or four. In the first case, the equation \( S = 0 \) has no positive roots, in the third, it has six; and in both cases the last term is positive. In the second case, the same equation has only one real and positive root, and the last term is negative.

The dimensions of \( S = 0 \), equal to \( \frac{4 \times 3}{2} \), being even, the function \( K^{(4)} \) will be positive in the first and third cases, and negative in the second case. Wherefore the proposed biquadratic equation will have only two real roots when the function \( K^{(4)} \), that is,

\[ (x' - x'')^2, (x' - x''')^2, (x' - x''''^2, (x'' - x''''^2, (x''' - x''''^2), \]

or \( 256r^3 - 128p^2r^2 + 144q^2pr + 16p^4r - 27q^4 - 4q^2p^2 \), is negative; and when the same function is positive, the proposed equation will have four real roots, if the terms of the auxiliary equation \( S = 0 \) be alternately positive and negative; otherwise it will have no real roots.

In an equation of the fifth degree, \( m \) is equal to one, or three, or five. In the first and third cases, the last term of \( S = 0 \) will be positive, for there are either no positive roots or ten; in the second case the last term is negative, the number of positive roots being three. The dimensions of \( S = 0 \), equal to \( \frac{5 \times 4}{2} \), being even, the function \( K^{(5)} \) will be positive in the first and third cases, and negative in the second. Wherefore the given equation of the fifth degree will have three real roots when the function \( K^{(5)} \) is negative; and when the same function is positive, it will have five real roots, if the terms of the auxiliary equation \( S = 0 \) be alternately positive and negative; otherwise it will have but one.

Resolution of Algebraic Equations.

16. When the coefficients of an equation are given in numbers, we may investigate the numerical value of any one root separately, by first seeking the limits between which it lies, and then narrowing those limits to any required degree of approximation. But this process is not what is meant by the general solution of algebraical equations, which supposes that the coefficients are denoted by general symbols, and consists in finding such a function of those quantities as shall, by the multiplicity of its values, represent all the roots. An algebraic expression is susceptible of many values, by means of the different radical quantities it contains; but, these radical quantities being themselves the roots of an equation, it follows that the general formula for the solution of any proposed equation can be nothing more than a function of the given coefficients combined with the roots of another equation.

The solution of quadratic equations has been known since the origin of algebra; it is found in the work of Diophantus, the first treatise on the science extant, if it be not the very first that was written. The Italian mathematicians, who are the founders of the modern algebra, discovered the solution of cubic and biquadratic equations. The rules they invented for this purpose are, however, rather the result of particular artifices, than deductions from any profound views of the structure of the equations they considered. In the course of the last and the present centuries, the general solution of equations has been the subject of almost innumerable researches by all the mathematicians of the first rank; but their labours have not been successful in advancing this branch of the science-beyond the steps made by the first algebraists.

The rules usually given for the solution of cubic and biquadratic equations are to be found in all the elementary books, and it would be superfluous to repeat them here. An account of the attempts that have been made to obtain a general theory for solving algebraic equations would greatly exceed the limits we must prescribe to ourselves. What has most impeded the progress of algebraists in their researches on this subject, is the difficulty of treating it by a perfect analysis, or of arriving at general conclusions by a process of reasoning founded solely on the principles of the inquiry, and disengaged from particular artifices of calculation, and from particular suppositions. In what follows, we shall endeavour to lay before our readers the general principles on which is founded all that has been successfully accomplished in this theory. Let the three roots of a cubic equation be represented by \(a\), \(b\), \(c\); and having interchanged these letters among one another, in all possible ways, we shall get the six permutations following, viz.

\[ abc,\ cab,\ bca \\ acb,\ bac,\ cba. \]

The combinations that stand first on the left are formed by prefixing the same letter to the permutations made with the other two; and those on each line are derived from one another by making the last letter of one stand first in that which follows, while the other two letters preserve the same order.

Now let \(z^3-1=0\); and let the letters of first combination of each line be prefixed in order to the three terms of \(1+z+z^2\); then we shall get

\[ t=a+bz+c_{z^2}, \quad s=a+c_{z}+b_{z^2}; \]

and if we multiply \(t\) and \(s\) by \(1\), \(z\), \(z^2\) successively, we shall further obtain

\[ t=a+b_{z}+c_{z^2}, \qquad s=a+c_{z}+b_{z^2} \\ t_{z}=c+a_{z}+b_{z^2}, \qquad s_{z}=b+a_{z}+c_{z^2} \\ t_{z^2}=b+c_{z}+a_{z^2}, \qquad s_{z^2}=c+b_{z}+a_{z^2}. \]

The six quantities \(t\), \(t_{z}\), \(t_{z^2}\), \(s\), \(s_{z}\), \(s_{z^2}\) comprehend all the values that can be formed by combining with \(1+z+z^2\), the three letters taken in any order whatever; and it is obvious that the cubes of all these six quantities, being each equal either to \(t^3\) or \(s^3\), have no more than two values.

And because \(t^3\) and \(s^3\) have only one value each, any symmetrical functions of them, as \(t^3+s^3\) and \(t^3s^3\), will have determinate values, which remain the same, however the letters \(a\), \(b\), \(c\) be interchanged among one another. The quantities \(t^3+s^3\) and \(t^3s^3\) must, therefore, be symmetrical functions of \(a\), \(b\), \(c\); and, consequently, they can be found in terms of the coefficients of the given equation.

By actually involving to the third power, we get

\[ t^3=a^3+b^3+c^3+6abc \\ +3(a^2b+b^2c+c^2a)\cdot z \\ +3(a^2c+c^2b+b^2a)\cdot z^2 \\ s^3=a^3+b^3+c^3+6abc \\ +3(a^2c+c^2b+b^2a)\cdot z \\ +3(a^2b+b^2c+c^2a)\cdot z^2. \]

and likewise

\[ (a+b+c)^3=a^3+b^3+c^3+6abc \\ +3(a^2b+b^2c+c^2a) \\ +3(a^2c+c^2b+b^2a). \]

Now \(1+z+z^2=0\), when \(z\) is any root of \(z^3-1=0\) different from unit; therefore, by adding the three last expressions, we get

\[ t^3+s^3=3(a^3+b^3+c^3)+18abc \\ -(a+b+c)^3. \]

Again, by actually multiplying

\[ ts=a^2+b^2+c^2 \\ +(ab+bc+ca)\cdot z \\ +(ab+bc+ca)\cdot z^2; \]

and, because \(z+z^2=-1\),

\[ ts=a^2+b^2+c^2 \\ -(ab+bc+ca). \]

By means of the preceding formulae, we can compute the values of \(t^3+s^3\) and \(t^3s^3\); and these values being the coefficients of a quadratic equation having its roots equal to \(t^3\) and \(s^3\), we can thence find \(t^3\) and \(s^3\), and \(t\) and \(s\). Now \(t\) and \(s\) being known, we have

\[ a+b+c=a+b+c \\ t=a+b_{z}+c_{z^2} \\ s=a+c_{z}+b_{z^2}; \]

wherefore,

\[ a=\frac{1}{3}(a+b+c)+\frac{1}{3}(t+s) \\ b=\frac{1}{3}(a+b+c)+\frac{1}{3}(t_{z}+s_{z}) \\ c=\frac{1}{3}(a+b+c)+\frac{1}{3}(t_{z^2}+s_{z^2}). \]

To apply the foregoing investigation, we shall take a cubic equation, \(x^3-3px-2q=0\), which is so prepared as to want the second term, then (Sect. 9)

\[ a+b+c=0 \\ ab+ac+bc=-3p \\ a^2+b^2+c^2=6p \\ a^3+b^3+c^3=6q \\ abc=2q. \]

consequently \(t^3+s^3=3^3\times2q\); \(ts=9p\), and \(t^3s^3=3^3\times3^3\cdot p^3\). Hence

\[ \frac{1}{3}t=(q+\sqrt{q^2-p^3})^{\frac{1}{3}} \\ \frac{1}{3}s=(q-\sqrt{q^2-p^3})^{\frac{1}{3}}. \]

Wherefore, by substituting these values in the expressions of the roots, we get

\[ a=(q+\sqrt{q^2-p^3})^{\frac{1}{3}}+(q-\sqrt{q^2-p^3})^{\frac{1}{3}} \\ b=z^2\cdot(q+\sqrt{q^2-p^3})^{\frac{1}{3}}+z\cdot(q-\sqrt{q^2-p^3})^{\frac{1}{3}} \\ c=z\cdot(q+\sqrt{q^2-p^3})^{\frac{1}{3}}+z^2\cdot(q-\sqrt{q^2-p^3})^{\frac{1}{3}}. \]

The preceding investigation, as well as all other methods that have been proposed for cubic equations, leads to the same result with the rule invented by Cardan; and, like that rule, it becomes, in some cases, insufficient for arithmetical computation, on account of the imaginary quantities that appear in the expressions of the roots. What is now mentioned is not an accidental circumstance, but a necessary consequence of the method of investigation pursued, and of the introduction of the imaginary roots of the equation \(z^3-1=0\). When \(a\), \(b\), \(c\), are real quantities, the values of \(t\) and \(s\) will be both imaginary, because they involve \(z\) and \(z^2\), or \(\frac{-1+\sqrt{-3}}{2}\) and \(\frac{-1-\sqrt{-3}}{2}\). In this case, therefore, although the three roots of the proposed equation are all real, yet the algebraic expressions of them are all imaginary, Equations, and useless for the purpose of numerical calculation; and the former circumstance is precisely the reason of the latter. On the other hand, when one root \(a\) is real and the other two imaginary, the impossible quantities destroy one another in the expressions of \(t\) and \(s\), which are, therefore, real quantities; and in this case, the algebraic formulae answer for finding the numerical values of the roots. The distinction here pointed out depends on the radical \(\sqrt{q^2-p^5}\), which is real or imaginary, according as the equation has one or three real roots, because \(q^2-p^5\) is always positive in the first case, and negative in the second.

Much labour and thought have been bestowed in order to free the formulae for the roots of cubic equations, from the imaginary expressions that render them unfit for arithmetical computation. In particular instances the difficulty disappears; namely, when the radical quantities are perfect cubes, in which cases the impossible parts of the cube roots destroy one another, so as to leave none but real quantities in the expressions of the roots of the equation. And by expanding the radical quantities we may, in all cases, obtain the roots of a cubic equation in series of an infinite number of terms free from the imaginary sign. But when it is required to transform the formulae for the case of a cubic equation with three real roots, into finite expressions free from impossible quantities, and to do so without employing any other than the received notations of algebra, all attempts to solve the problem have led to equations in the same circumstances with the one proposed, and have ended in bringing back the same difficulty; in so much that equations of the description mentioned are said to be in the irreducible case.

It is, however, possible to transform the formulae for the roots of a cubic equation in the irreducible case into real expressions, although not so as to fulfil all the conditions above mentioned. Let \(q^2-p^5=y^2\); then \(p=(q^2-y^2)^{\frac{1}{3}}\); wherefore the equation \(x^3-3px-2q=0\), will become

\[ x^3-3(q^2-y^2)^{\frac{1}{3}}x-2q=0 \quad (1). \]

By the preceding formula, the value of \(x\) in this equation will be

\[ x=(q+y)^{\frac{1}{3}}+(q-y)^{\frac{1}{3}}, \]

or, according to the notation of Sect. 10, making \(n=\frac{1}{3}\),

\[ x=2H_{\frac{1}{3}}(q, y^2). \]

By substituting this value of \(x\) we get

\[ \left\{ 2H_{\frac{1}{3}}(q, y^2) \right\}^3-3(q^2-y^2)^{\frac{1}{3}} \left\{ 2H_{\frac{1}{3}}(q, y^2) \right\}-2q=0; \]

which equation being true for all values of \(q\) and \(y^2\), must be identical, or, when expanded, must consist of a series of quantities that mutually destroy one another. Now the equation will still be identical, when \(y^2\) is changed into \(-y^2\); so that we shall have

\[ \left\{ 2H_{\frac{1}{3}}(q, -y^2) \right\}^3-3(q^2+y^2)^{\frac{1}{3}} \left\{ 2H_{\frac{1}{3}}(q, -y^2) \right\}-2q=0; \]

and this proves that the equation

\[ x^3-3(q^2+y^2)^{\frac{1}{3}}x-2q=0 \quad (2) \]

is solved by the formula

\[ x=2H_{\frac{1}{3}}(q, -y^2). \]

As the investigation in Sect. 10 is equally true, whether \(n\) be a whole or a fractional number, we may apply it to find the value of the symbol \(2H_{\frac{1}{3}}(q, -y^2)\).

For this purpose, let

\[ q=r \cos \varphi = r \cos (\varphi + 360^\circ) = r \cos (\varphi + 2.360^\circ), \] \[ y=r \sin \varphi = r \sin (\varphi + 360^\circ) = r \sin (\varphi + 2.360^\circ); \]

then \(r=\sqrt{q^2+y^2}\); and, according as we take one or other of the angles that have the same sines and cosines, we shall obtain three different values of \(2H_{\frac{1}{3}}(q, -y^2)\), or of \(x\), viz.

\[ a=2r^{\frac{1}{3}} \cdot \cos \frac{\varphi}{3} \] \[ b=2r^{\frac{1}{3}} \cdot \cos \left( \frac{\varphi}{3} + 120^\circ \right) \] \[ c=2r^{\frac{1}{3}} \cdot \cos \left( \frac{\varphi}{3} + 240^\circ \right). \]

By putting \(p=(q^2+y^2)^{\frac{1}{3}}\), the equation (2) will assume the same form as at first, namely,

\[ x^3-3px-2q=0; \]

and because \(p^3=q^2+y^2=r^2\), and \(y=\sqrt{p^2-q^2}\); if we determine the angles by means of their tangents, instead of their sines and cosines, we shall get

\[ \frac{\sqrt{p^2-q^2}}{q} = \tan \varphi = \tan (\varphi + 360^\circ) = \tan (\varphi + 2 \cdot 360^\circ); \]

and the three roots of the equation will be

\[ a=2\sqrt{p} \cdot \cos \frac{\varphi}{3} \] \[ b=2\sqrt{p} \cdot \cos \left( \frac{\varphi}{3} + 120^\circ \right) \] \[ c=2\sqrt{p} \cdot \cos \left( \frac{\varphi}{3} + 240^\circ \right). \]

Every cubic equation falls under one or other of the formulae (1) and (2), except when \(y=0\), or \(p^3=q^2\), which takes place when an equation changes from one class to another; and in this case we have

\[ x^2-3q^{\frac{2}{3}}x-2q=(x-2q^{\frac{1}{3}})(x+q^{\frac{1}{3}})(x+q^{\frac{1}{3}}). \]

The several rules that have now been given, therefore, include every possible case.

The difficulty attending the irreducible case arises from a real distinction between the two subordinate classes of cubic equations, and is insurmountable by the ordinary operations of algebra. There is no permanent distinctions of equations belonging to the same order, when we consider their roots as positive or negative; because, in any proposed equation, all the roots, or as many of them as we please, can be changed from positive to negative, by the simple artifice of increasing or diminishing them all by a given quantity. But the case is otherwise when we consider the roots of an equation in their character of real or imaginary quantities. No transformation can change an equation with one real root into another with three real roots, without involving the operations of the impossible arithmetic. If, therefore, we lay down this condition, namely, that the formulae for the roots of equations must be in a shape fit for numerical calculation; we may conclude that, in fact, there is no resolution of equations except what consists in reducing all those of the same class to some one of that class, the most simple and convenient in its form, that can be found. If we examine the preceding investigation, it will appear that it is merely an attempt to reduce all cubic equations to the form \( x^3 - A = 0 \); and this readily succeeds, without impossible operations, when the proposed equation and that with which it is compared have their roots of a similar description; and it as surely fails when the case is otherwise.

In geometry, where the relations of the magnitudes under consideration are never lost sight of, there is no tendency to refer the solution of a problem to a class to which it does not belong. The ancient geometer could never be in danger of applying the problem for finding two mean proportionals to a case that can be constructed only by the trisection of an angle. The modern analyst, dismissing the original magnitudes of his problem, and reducing all possible relations to equations in abstract numbers, is apt to overlook distinctions, and sometimes to waste his labour, in seeking to accomplish what a due separation of cases would show to be impossible. There is the same distinction between the class of cubic equations with one real root, and that with three real roots, that there is between the two geometrical problems alluded to above; and the algebraist who attempts, by means of the ordinary operations of his art, to transform Cardan's formula so as to make it apply to the irreducible case, is precisely in the same situation with the geometer who should set about trisecting an angle by finding two mean proportionals.

The power and force of the algebraic method does not consist in breaking down real distinctions, but in connecting, by sure and general principles, many truths which, in geometry, are joined only by vague analogies, and even have no affinity at all. This advantage is derived chiefly from the doctrine of negative quantities, and from the impossible arithmetic. By means of the first, a formula which is obtained by considering only one state of the data of a problem, applies, necessarily and by the very structure of analytical language, to the same problem in all possible conditions of the data. On the other hand, when the relations of the data vary, the geometer is obliged to subdivide his problem into cases, or into other subordinate problems; and although it may be perceived that great similitude prevails among all the subdivisions, yet it is impossible to reduce the analogy between them to determinate rules, as is done in algebra. But, in the whole compass of geometry, there is nothing that bears any resemblance to the imaginary arithmetic. When the geometer has fixed the 'determination' of his problem, or ascertained the limits within which it is possible, he has drawn a line that must be the boundary of his investigation. Now, it is to truths lying beyond this line that the meaning of the comprehensive expressions of the imaginary arithmetic must be referred. It is not to be understood that a problem can be solved by algebra, which is impossible in geometry; but the analytical formulae, at the same time that they mark the limits of the problem, go beyond them, and point out connected truths, that require only certain changes to be made in the algebraic expressions; in like manner, as all the possible cases of the same problem are derived from one only, by means of the variations of the signs.

If \( a, b, c, d \), represent the four roots of a biquadratic equation; and if we prefix the same letter \( a \) to all the permutations made with the other three, we shall get the six combinations following, viz.

\[ abcd,\quad abdc,\quad acdb,\quad adcb,\quad acbd,\quad abdc. \]

In the first line, the letters \( b, c, d \), are made to circulate, by placing immediately after the immovable letter \( a \) that which stands last in the combination preceding; and, in the second line, the movable letters have, respectively, an inverted order to what they have in the first line.

Let \( \varepsilon^2 - 1 = 0 \); and let the four letters taken in the several orders of the six combinations be prefixed to the terms of \( 1 + \varepsilon + \varepsilon^2 + \varepsilon^3 \); the results of the first line being \( t, t', t'' \), and those of the second line \( s, s', s'' \); then

\[ \begin{align*} t &= a + b\varepsilon + c\varepsilon^2 + d\varepsilon^3 \\ t' &= a + d\varepsilon + b\varepsilon^2 + c\varepsilon^3 \\ t'' &= a + c\varepsilon + d\varepsilon^2 + b\varepsilon^3, \end{align*} \] \[ \begin{align*} s &= a + d\varepsilon + c\varepsilon^2 + b\varepsilon^3 \\ s' &= a + c\varepsilon + b\varepsilon^2 + d\varepsilon^3 \\ s'' &= a + b\varepsilon + d\varepsilon^2 + c\varepsilon^3. \end{align*} \]

Now, in the equation \( \varepsilon^2 - 1 = 0 \), \( \varepsilon \) is either equal to \( +1 \), or to \( -1 \); and whether we take the one value or the other, it is apparent that \( t = s, t' = s', t'' = s'' \).

Again, from every one of the six foregoing combinations, four others are derived by circulating the letters continually from the last place to the first; and, in this manner, we obtain twenty-four different permutations, which are all that can be made with four letters. Thus, if we take \( abcd \), and move the letters as directed, we shall get these four combinations, viz.

\[ abcd,\quad dabc,\quad cdab,\quad bcda. \]

And if we multiply \( t \) by \( \varepsilon \) continually, observing to retain the three first powers of \( \varepsilon \), and to make \( \varepsilon^4 = 1 \), we shall get

\[ \begin{align*} t &= a + b\varepsilon + c\varepsilon^2 + d\varepsilon^3 \\ t\varepsilon &= d + a\varepsilon + b\varepsilon^2 + c\varepsilon^3 \\ t\varepsilon^2 &= c + d\varepsilon + a\varepsilon^2 + b\varepsilon^3 \\ t\varepsilon^3 &= b + c\varepsilon + d\varepsilon^2 + a\varepsilon^3; \end{align*} \] Equations, so that \( t, t_2, t_2^2, t_2^3 \), are the functions formed by prefixing to \( 1 + \varepsilon + \varepsilon^2 + \varepsilon^3 \), the letters of the four combinations; and it is obvious that these functions have all the same square, equal to \( t^2 \).

Wherefore, if the four letters, taken in all possible orders, be prefixed to the terms of \( 1 + \varepsilon + \varepsilon^2 + \varepsilon^3 \), the squares of the twenty-four resulting functions will be equal to one or other of the six quantities, \( t^2, t'^2, t''^2, s^2, s'^2, s''^2 \); and since it has been proved that \( t = s, t' = s', t'' = s'' \); it follows that the twenty-four squares have no more than three different values, equal to \( t^2, t'^2, t''^2 \).

And, because \( t^2, t'^2, t''^2 \), can have no more than one value each, any symmetrical functions of them, viz.

\[ t^2 + t'^2 + t''^2 \\ t^2 t'^2 + t'^2 t''^2 + t''^2 t^2 \\ t^2 t'^2 t''^2, \]

will have determinate values independent of the order of the letters \( a, b, c, d \). The same functions will therefore be symmetrical expressions of the roots of the given biquadratic equation, and they will be known in terms of the coefficients of that equation.

Supposing \( \varepsilon = -1 \), we get

\[ t = a - b + c - d \\ t' = a - d + b - c \\ t'' = a - c + d - b ; \]

and hence,

\[ t^2 = a^2 + b^2 + c^2 + d^2 \\ - 2(ab + ad + bc + cd) + 2(ac + bd) \\ = (a + b + c + d)^2 - 4 \Sigma ab + 4(ac + bd); \]

the symbol \( \Sigma ab \) being used here, as in Sect. 9, to denote the sum of the products of every two of the roots. Wherefore, if we put

\[ M = (a + b + c + d)^2 - 4 \Sigma ab \\ m = ac + bd \\ m' = ab + dc \\ m'' = ad + bc, \]

then

\[ t^2 = M + 4m \\ t'^2 = M + 4m' \\ t''^2 = M + 4m'' ; \]

and hence,

\[ t^2 + t'^2 + t''^2 = 3M + 4(m + m' + m'') \\ t^2 t'^2 + t'^2 t''^2 + t''^2 t^2 = 3M^2 + 8M(m + m' + m'') + 16(mm' + mm'' + m'm''). \]

But it will readily appear that

\[ m + m' + m'' = \Sigma ab \\ mm' + mm'' + m'm'' = (a + b + c + d) \times \Sigma abc \\ - 4abcd. \]

Now, by substituting these values, we get

\[ t^2 + t'^2 + t''^2 = 3(a + b + c + d)^2 - 8 \Sigma ab \\ t^2 t'^2 + t'^2 t''^2 + t''^2 t^2 = 3(a + b + c + d)^4 - 16(a + b + c + d)^2 \times \Sigma ab \\ + 16(a + b + c + d) \times \Sigma abc + 16(\Sigma ab)^2 \\ - 64abcd. \]

Again, if we multiply the expressions of \( t, t', t'' \), Equations. we shall get

\[ tt' = (a - c)(a^2 - c^2) + (b - d)(b^2 - d^2) \\ - (a + c)(b - d)^2 - (b + d)(a - c)^2; \]

or,

\[ tt'' = a^3 + b^3 + c^3 + d^3 + 2\Sigma abc \\ - (a^2b + a^2c + a^2d + b^2a + b^2c + b^2d \\ + c^2a + c^2b + c^2d + d^2a + d^2b + d^2c); \]

and finally, by means of the formulae in Sect. 9,

\[ tt'' = (a + b + c + d)^3 + 8\Sigma abc \\ - 4(a + b + c + d) \times \Sigma ab. \]

If now we substitute the values computed by the preceding formulae, in the cubic equation,

\[ o = u^3 - (t^2 + t'^2 + t''^2)u^2 \\ + (t^2 t'^2 + t'^2 t''^2 + t''^2 t^2)u \\ - t^2 t'^2 t''^2, \]

we shall obtain the values of \( t^2, t'^2, t''^2 \), and consequently of \( t, t', t'' \), by solving that equation: and, when \( t, t', t'' \) are known, we have

\[ a + b + c + d = a + b + c + d \\ t = a + b_2 + c_2 + d_2 \\ t' = a + d_2 + b_2 + c_2 \\ t'' = a + c_2 + d_2 + b_2; \]

wherefore, because \( o = 1 + \varepsilon + \varepsilon^2 + \varepsilon^3 \), we get

\[ a = \frac{1}{4} \left\{ a + b + c + d + t + t' + t'' \right\} \\ b = \frac{1}{4} \left\{ a + b + c + d + t_2 + t'_2 + t''_2 \right\} \\ c = \frac{1}{4} \left\{ a + b + c + d + t_3 + t'_3 + t''_3 \right\} \\ d = \frac{1}{4} \left\{ a + b + c + d + t_4 + t'_4 + t''_4 \right\} \]

And finally, by making \( \varepsilon = -1 \),

\[ a = \frac{1}{4} \left\{ a + b + c + d + t + t' + t'' \right\} \\ b = \frac{1}{4} \left\{ a + b + c + d - t - t' - t'' \right\} \\ c = \frac{1}{4} \left\{ a + b + c + d - t' - t'' - t \right\} \\ d = \frac{1}{4} \left\{ a + b + c + d - t - t' - t'' \right\}. \]

In applying these formulae, it is necessary to observe, that, as the quantities \( t, t', t'' \), are found by extracting the square root, they may each have either the sign plus or the sign minus prefixed. But all ambiguity from this cause will be taken away, if it be observed, that the expressions of \( a, b, c, d \), will always give the same results, provided the signs of \( t, t', t'' \), be so determined as to satisfy the equation,

\[ tt'' = (a + b + c + d)^2 + 8 \Sigma abc \\ - 4(a + b + c + d) \times \Sigma ab. \]

For, if we suppose that the signs of \( t, t', t'' \), are so determined as to satisfy the equation mentioned, they cannot be varied so as still to satisfy the same equation, unless two of them be changed together; for, if one sign only be changed, or if all the three be changed together, the product \(tt't''\) will have an opposite sign to what it had before, and the equation will no longer be satisfied. But the expressions of \(a, b, c, d\), give the same set of values when the signs of any two of the letters \(t, t', t''\), are changed together; so that, in order to have the true values of the quantities sought, no other rule for the signs of \(t, t', t''\), is necessary than that they must be such as to satisfy the equations alluded to.

To apply the preceding investigation we may take the equation,

\[ x^4 + px^2 + qx + r = 0, \]

which wants the second term. Then,

\[ \begin{align*} o &= -a + b + c + d \\ p &= \Sigma ab \\ -q &= \Sigma abc \\ r &= abcd : \end{align*} \]

hence

\[ \begin{align*} t^2 + t'^2 + t''^2 &= -8p \\ t^2 t'^2 + t^2 t''^2 + t'^2 t''^2 &= 16p^2 - 64r \\ tt' t'' &= -8q ; \end{align*} \]

and \(t^2, t'^2, t''^2\), are the roots of the cubic equation,

\[ u^3 + 8pu^2 + 16(p^2 - 4r)u - 64q^2 = 0. \]

Having solved this equation, and found the values of \(t, t', t''\), the signs of these quantities must next be determined so as to satisfy the equation,

\[ tt' t'' = -8q ; \]

and then we have these formulae for computing the roots of the proposed equation, viz.

\[ \begin{align*} a &= \frac{t + t' + t''}{4} \\ b &= \frac{-t + t' - t''}{4} \\ c &= \frac{t - t' - t''}{4} \\ d &= \frac{-t - t' + t''}{4} \end{align*} \]

These formulae coincide with the method of solving biquadratic equations first proposed by Euler in his Algebra. But, in order to take away the ambiguity arising from the double sign of the square root, that celebrated mathematician uses two sets of expressions for the roots of the equation, viz.

\[ \begin{align*} a &= \frac{t + t' + t''}{4} & a &= \frac{-t - t' - t''}{4} \\ b &= \frac{-t + t' - t''}{4} & b &= \frac{t - t' + t''}{4} \\ c &= \frac{t - t' - t''}{4} & c &= \frac{-t + t' + t''}{4} \\ d &= \frac{-t - t' + t''}{4} & d &= \frac{t + t' - t''}{4} \end{align*} \]

of which one set is the same with the formulae given above, and the other is obtained by changing \(t, t', t''\) into \(-t, -t', -t''\); the first set being directed to be used when \(-8q\) is positive, and the other set when the same quantity is negative. This procedure is not so simple as that we have followed, which requires only one set of formulae. It has even been the occasion of leading into error, in as much as it makes the signs of \(t, t', t''\), depend entirely upon the sign of the given quantity \(-8q\); whereas, it is indispensable that, regard being had to the nature of the quantities \(t, t', t''\), their signs shall be determined so as to satisfy the equation \(tt' t'' = -8q\). This inadvertence of Euler has escaped the observation of most of the authors who have treated of biquadratic equations, and was first noticed by M. Bret in the second volume of the Correspondance sur l'Ecole Polytechnique.

It may not be improper to notice briefly some of the other rules for biquadratic equations. These are chiefly two; the method of Descartes, which resolves the given equation into two quadratic factors; and the oldest method of all, invented by Louis Ferrari, a pupil of Cardan, which proceeds by transforming the given equation, so as to make it equal to the difference of two complete squares, and then extracting the square roots. However different from one another these two methods may at first seem, they are at bottom the same; and they are so far connected with that already investigated, that all the three lead to the same cubic equation.

Suppose that \(a, b, c, d\), are the roots of the biquadratic equation,

\[ x^4 - Ax^2 + Bx^2 - Cx + D = 0 ; \]

then \(x^2 - (a+b)x + ab = 0\), and \(x^2 - (c+d)x + cd = 0\), are two quadratic factors, the product of which is equal to the given equation. Now,

\[ \begin{align*} A &= a + b + c + d \\ t &= a + b - c - d ; \end{align*} \]

wherefore, if we put \(ab = p + y, cd = p - y\), the two factors will become

\[ \begin{align*} x^2 - \frac{1}{2}(A + t)x + p + y &= 0 \\ x^2 - \frac{1}{2}(A - t)x + p - y &= 0 ; \end{align*} \]

and if we multiply them, and equate the coefficients of the product to the coefficients of the given equation, we shall get

\[ \begin{align*} 2p + \frac{1}{4}A^2 - \frac{1}{4}t^2 &= B \\ Ap + ty &= C \\ y^2 - y^2 &= D . \end{align*} \]

And it is to be observed that, on account of the two first of these equations, \(p\) and \(y\) are both real quantities when \(t\) is a real quantity; so that, provided a real value of \(t\) can be found, the given equation is always resolved, by this method, into two quadratic factors free from imaginary expressions.

Now, by combining the equations just found, we shall get

\[ \begin{align*} o &= t^6 - (3A^2 - 8B)t^4 \\ &\quad + (3A^4 - 16A^2B + 16B^2 + 16AC - 64D)t^2 \\ &\quad - (A^5 - 4AB + 8C)^2 , \end{align*} \] p = \frac{1}{2}B - \frac{1}{8}A^2 + \frac{1}{8}t^2, \[ y = \sqrt{\left( \frac{1}{2}B - \frac{1}{8}A^2 + \frac{1}{8}t^2 \right)^2 - D}. \]

The first of these equations is a cubic, of which the root is \( t^2 \); and it is precisely the same with the cubic of the former method. As the last term of this equation is essentially positive, it follows, that there is always one positive value of \( t^2 \), and one real value of \( t \); wherefore, in consequence of what has been proved, the values of \( p \) and \( y \), derived from the positive value of \( t^2 \), are in every case real quantities, which is, no doubt, an advantage in the practical application of the method.

If we wish to follow the process of Louis Ferrari, we may assume \( p, t, y \), so as to render the expression

\[ \left( x^2 - \frac{1}{2}Ax + p \right)^2 - \left( \frac{1}{2}tx + y \right)^2 = 0, \]

identical with the given equation; and as this expression is no more than the product of the two quadratic factors of the last method, the quantities to be determined will be found by the formula already given.

The theory of permutations, which is successful in solving cubic and biquadratic equations, applies likewise to those of the fifth and higher orders. But, to use the words of Lagrange, "Passé le quatrième degré, la méthode, quoiqu' applicable en général, ne conduit plus qu'à des équations résolvantes de degrés supérieurs à celui de la proposée." Thus, in the case of equations of the fifth degree, the theory leads to a biquadratic equation of which the coefficients are to be found by resolving an equation of the sixth order.

There is, however, no doubt that the doctrine of permutations contains the principles from which we are to expect the resolution of equations of the higher orders, if the problem be possible. It may be alleged, with great probability, that the theory succeeds in the less complicated cases, because, when the number of the roots is small, their permutations are soon exhausted, and we speedily arrive at those combinations of them which remain invariable, whatever be the order of the quantities combined. But when the number of the roots is greater than four, their permutations are very numerous, and, at the same time, the functions produced by combining them are very complicated; on which accounts it is difficult to conduct the investigation so as to arrive at a satisfactory conclusion, either accomplishing the intended purpose, or proving that the undertaking is impossible.

In the twelfth volume of the Italian Society, and in a work published at Modena in 1813, M. Paolo Ruffini has proved, that no function of five letters can exist that is susceptible of only three or four different values when the letters are interchanged among one another in all possible ways. M. Cauchy, in the sixteenth volume of the Journal de l'Ecole Polytechnique, has demonstrated, that a function of \( n \) letters, unless it have no more than two different values, cannot have a number of different values less than the prime number next below \( n \). On these grounds, it has been inferred, that the resolution of equations of the fifth degree is in reality an impossible problem. (Lacroix, Compt. des Elem. d'Algebrec, p. 61.) And, if it be admitted that, in the process of resolution, no equations can occur except such as have symmetrical functions of the five letters for their coefficients, the inference founded on the labours of the eminent mathematicians we have mentioned would be indisputable. But it is not impossible that the resolution of equations of a high order must be effected by gradually depressing an equation at first of great dimensions; and in this procedure we may arrive at equations, the coefficients of which, although functions of the roots of the proposed equation, are not symmetrical functions, but partial expressions susceptible of several values, according as the order of the letters that denote the roots is made to vary. On this supposition, the resolution of equations above the fourth order, by means of equations inferior in degree, would not be inconsistent with what has been proved.

17. A method for solving equations of one order may be generalized so as to extend to a certain class in all orders. Thus De Moivre has found a species of equations of every degree that have their roots similar to those of cubics, and which are solved by the formula

\[ x = \frac{1}{(q + \sqrt{q^2 - p^n})^n} + \frac{1}{(q - \sqrt{q^2 - p^n})^n}, \]

differing in no respect from the expression for resolving cubics, except that \( n \) is written in place of 3.

An equation may be depressed to a lower order when it is known that the roots have a given relation to one another. An instance of this has already occurred in the case of equal roots; for, the equal roots having been first found, the equation can be lowered by division. Reciprocal equations furnish another example of depression to a lower order, on account of a relation subsisting among the roots. A reciprocal equation is one of even dimensions, such that half the roots are respectively the reciprocals of the other half, in which case no alteration is produced in the equation when \( \frac{1}{x} \) is substituted for \( x \). In equations of this kind, the same coefficients occur in the same order, and with the same signs, reckoning from either end; a description that likewise applies to some equations of odd dimensions, which, however, do not constitute a new class, being merely reciprocal equations, as defined above, multiplied by the factor \( x+1 \). A reciprocal equation may always be depressed to half the dimensions, by transforming it so that the new unknown quantity shall be equal to \( x + \frac{1}{x} \). It is sufficient to have mentioned these cases, which are fully treated of in all the elementary books.

Equations with only two terms, as \( x^n - 1 = 0 \), are the most extensive class that have been resolved by a general method. The successful application of analysis to this class of equations is extremely interest- Equations. ing both in itself, and likewise because it is connected with the division of the circle into equal parts, and has occasioned the discovery of some curious, and unexpected results respecting that problem. For these reasons, it appears proper to lay before our readers a short view of this branch of the doctrine of algebraic equations.

We have already shown, that, admitting the theory of angular sections, every equation with only two terms, as \( x^p - 1 = 0 \), may be completely resolved into its binomial and trinomial factors; and hence all its roots, possible and impossible, may be computed by means of the trigonometrical tables in common use. If we put \( \varphi = \frac{6p^{90}}{3} \), and denote by \( k \) any number less than \( \frac{1}{2} p \), we have found that the equation \( x^p - 1 = 0 \) is divisible by the quadratic factor \( x^2 - 2x \cos \varphi + 1 \), and, consequently, that it has the two impossible roots,

\[ x = \cos k \varphi + \sin k \varphi \cdot \sqrt{-1} \] \[ x = \cos k \varphi - \sin k \varphi \cdot \sqrt{-1} \]

and, because \( \cos k \varphi = \cos (p - k) \varphi \), and \( -\sin k \varphi = \sin (p - k) \varphi \), the same two roots may be otherwise more symmetrically represented, thus,

\[ x = \cos k \varphi + \sin k \varphi \cdot \sqrt{-1} \] \[ x = \cos (p - k) \varphi + \sin (p - k) \varphi \cdot \sqrt{-1}. \]

Therefore, when \( p \) is odd, the equation \( x^p - 1 = 0 \) has one real root equal to 1; and when \( p \) is even, it has two real roots equal to \( \pm 1 \); and in both cases the remaining roots are all impossible, and are found from the formula,

\[ x = \cos k \varphi + \sin k \varphi \cdot \sqrt{-1}, \]

by making \( k \) equal to all the integral numbers less than \( p \) in the one case, and less than \( p - 1 \) in the other. Nothing, therefore, can be more simple than the computation of the roots of such equations by means of the trigonometrical tables. But in seeking a general solution, it is required to investigate the roots without resorting to the properties of the circle, unless in so far as this may be necessary for solving similar equations inferior in degree to the one proposed. In this view the resolution of the equation

\[ x^p - 1 = 0, \]

is equivalent to the division of the circle into \( p \) equal parts, granting the like division for all numbers less than \( p \). And in order to render the investigation of the problem as simple as possible, it may be further observed, that it will be sufficient to consider the case when the exponent is a prime number; because, from this case, the other, when it is a composite number, can be readily deduced.

It will be proper to premise here a property of the roots of equations with only two terms, to which we shall have occasion continually to refer. The property in question depends upon this theorem, namely: When \( k \) is any number, not a multiple of the prime number \( p \), the remainders of the terms of the series,

\[ 1 \times k, 2 \times k, 3 \times k \ldots (p-1) \times k, \]

when each is divided by \( p \), are all different from one another; and, consequently, without regard to the order, they will coincide with the numbers 1, 2, 3, &c. less than \( p \). If, therefore, we take any one of the impossible roots of the equation \( x^{p-1} \), viz.

\[ r = \cos k \varphi + \sin k \varphi \cdot \sqrt{-1}, \]

all its powers with indices less than \( p \), viz.

\[ r^2 = \cos 2k \varphi + \sin 2k \varphi \cdot \sqrt{-1}, \] \[ r^3 = \cos 3k \varphi + \sin 3k \varphi \cdot \sqrt{-1}, \] \[ \text{&c.} \]

will be different from one another; and likewise they will coincide, without regard to the order, with the like powers of any other impossible root of the same equation: because, whatever number \( k \) stands for, the arcs are all different from one another, and, neglecting whole circumferences, constitute the same series of terms although in different orders. Wherefore, \( p \) being a prime number, if \( r \) be one of the impossible roots of the equation \( x^p - 1 = 0 \), all the roots will be represented by the terms of the geometrical progression,

\[ r^0, r^1, r^2, r^3, \ldots, r^{p-1}; \]

for every one of these terms satisfy the given equation, and it has been shown that they are all different from one another.

When \( p \) is a composite number, the same property does not belong to all the roots of the equation \( x^p - 1 = 0 \), but only to some of them. It belongs generally to the root

\[ r = \cos k \varphi + \sin k \varphi \cdot \sqrt{-1}, \]

when \( k \) is either equal to unit, or to any number that has no common divisor with \( p \); in which cases, all the powers of \( r \) are roots of the equation \( x^p - 1 = 0 \), and all different from one another, when the exponents are different and less than \( p \).

If the equation \( x^p - 1 = 0 \) be divided by the binomial factor \( x - 1 \), we shall get

\[ x^{p-1} + x^{p-2} + x^{p-3} \ldots + x + 1 = 0; \]

and this being a reciprocal equation, it can be farther depressed to half the dimensions. In this manner we obtain the solution of \( x^7 - 1 = 0 \), which is reduced to a cubic; but, by the same procedure, the equation next in order, viz. \( x^{11} - 1 = 0 \), can be lowered only to the fifth degree, for equations of which class there is no rule. Nevertheless, this last equation has been solved by Vandermonde, to whom, and to Lagrange, we are mainly indebted for disengaging the resolution of equations from the complicated operations of algebra, and for substituting, in their place, reasonings founded on the doctrine of combinations. The author has not explained particularly the process by which his solution was obtained; he gives it as a result of his theory, which, although it fails in general for equations above the fourth degree, succeeds in this instance on account of particular relations between the roots. Similar relations subsist between the roots of any other binomial equation when the exponent is a prime number; and, in consequence, a like mode of investigation will apply, as indeed the author has expressly said. But this procedure would unavoidably be attended in every new instance with very long calculations; and it appears hardly possible to arrive in this way at any general method that would apply to all equations of the class in a regular manner, and without considerations drawn from each particular case.

M. Gauss, in a work entitled Disquisitiones Arithmeticae, replete with original and important matter, applied a property of prime numbers to the solution of binomial equations, which removed every difficulty, and led to a theory that unites simplicity and generality. If we suppose that \( p \) is a prime number, and resolve \( p-1 \) into its component factors, so that \[ p-1 = a^{\lambda} \cdot b^{\mu} \cdot c^{\gamma} \text{ &c., } a, b, c, \text{ &c. being prime numbers}, \] M. Gauss has proved that the solution of the equation \( x^{p}-1=0 \), or, which is the same thing, the division of the circle into \( p \) equal parts, can be effected by solving successively \( \lambda \) equations of \( a \) dimensions, \( \mu \) equations of \( b \) dimensions, \( \gamma \) equations of \( c \) dimensions, &c. Thus, if \( p=13 \), then, because \( 13-1=3 \times 2^{2} \), the roots of \( x^{15}-1=0 \) can be found, or a polygon of 13 sides can be inscribed in a circle, by solving a cubic and two quadratic equations in succession. In certain cases, when a prime number comes under the form \( 2^{n}+1 \), as 17, 257, &c., the division of the circle will require the solution of equations no higher than the second order; whence this unexpected consequence has resulted from the theory of M. Gauss, that the inscription of a polygon of 17, or 257 sides in a circle, which are problems that have always been understood to transcend the limits of the elementary geometry, can, nevertheless, be constructed by the operations admitted in that science.

A work replete with so many interesting discoveries as the Disquisitiones Arithmeticae, could not fail to excite the attention of mathematicians. Legendre, in republishing his Essay on the Theory of Numbers, has added to it an exposition of M. Gauss's theory of binomial equations; and the same theory is the subject of the 14th note in the second edition of Lagrange's Treatise on Numerical Equations. No part of the mathematics could pass through the hands of men of so much ability without receiving great improvement. Lagrange has shown, that it is not necessary to go through the several intermediate equations that make so essential a part in the investigation of M. Gauss; and, by this means, he has reduced the solution of equations with two terms to the utmost simplicity of which it is capable. But, in one respect, it must be admitted that the procedure of the illustrious geometer is imperfect. Although it arrives, by a short investigation, at the partial quantities that by their additions form the expressions of the roots sought, it leaves indeterminate the order in which they are to be combined. M. Gauss has avoided ambiguity in this respect by deducing from one of the quantities all the other parts of the same expression; but, amidst a multiplicity of different systems of values that may be deduced from the partial quantities, Lagrange has given no clue to guide to the true one.

In laying before our readers some account of this interesting branch of the theory of algebraic equations, we shall view the subject in a light somewhat different from that in which it has hitherto been placed. Instead of seeking directly the roots of binomial equations, we shall apply the principles of M. Gauss's theory immediately to the division of the circle into equal parts, by taking the arcs of the circumference in that order, to which the method owes all its success. This procedure is attended with some advantages. In the first place, the algebraic expressions of the quantities sought, represented by \[ \cos \frac{k \times 360^{\circ}}{p} \] are more simple than those of the imaginary roots of the corresponding binomial equation; and, in the second place, the same expressions, having always real values, are better fitted for application than the roots of binomial equations which require to be further reduced to prepare them for calculation.

Before entering on the principal problem, it is necessary to say something of that property of numbers on which the whole theory depends. Supposing \( p \) to be any prime number, Euler has distinguished by the name of a Primitive Root any number less than \( p-1 \), such that, if we take the series of all its powers with indices less than \( p \), and in each power reject the multiples of \( p \) it contains, the several remainders are all different from one another, and, consequently, paying no regard to the order, they will coincide with the numbers 1, 2, 3, &c. less than \( p \). It has been proved that, for every prime number, there are as many primitive roots as there are numbers less than \( p-1 \), which have no common divisor with it. The existence of such numbers in every case is therefore demonstrated; but no direct method of finding them has yet been published with which we are acquainted.

We gladly seize the present occasion of laying down a rule for finding the primitive roots of a prime number. But first we must premise, that when any proposed number is said to satisfy the equation \( x^{m}+1=0 \), it is always understood that the multiples of the prime number \( p \) are rejected; and the meaning is, that, when the given number is substituted for \( x \), the whole result is divisible by \( p \) without any remainder.

Now, let \( p \) be a prime number, and \( a, b, c, \) &c. the prime divisors of \( p-1 \), so that \( p-1=2^{\tau} \cdot a^{\lambda} \cdot b^{\mu} \cdot c^{\gamma} \). &c.: then every primitive root will satisfy the first of the following equations without satisfying any of the rest, viz.

\[ \frac{p-1}{x^{2}+1}=0 \]

\[ \frac{p-1}{x^{2a}+1}=0 \] \[ \frac{p-1}{x^{2b}} + 1 = 0 \\ \frac{p-1}{x^{2c}} + 1 = 0 \\ \text{&c.} \] And, on the other hand, every number, not a primitive root, which satisfies the first equation, will, at the same time, satisfy one, or more, or all, of the other equations.

But the numbers which satisfy the first equation are exclusively those which are not found among the remainders of the series of square numbers divided by \( p \). Wherefore, setting aside the first equation, if we seek among the non-residual numbers for such as satisfy none of the remaining equations, the numbers so found will be the primitive roots sought.

When one primitive root is found by this method, all the rest may be directly obtained from it. For, if \( 1, w, w', w'', \text{&c.} \ldots w^{(n)} \), represent all the numbers less than \( p-1 \) and prime to it; then, \( a \) being one of the primitive roots, all the roots will be equal to the series of powers, \[ a', a^w, a^{w'}, a^{w''}, \ldots, a^{w(n)} \] rejecting always the multiples of \( p \).

The demonstration of these properties would lead us aside from our present purpose; and we shall be content with adding some examples for the sake of illustration.

Let \( p=11 \); then \( \frac{p-1}{2} = 5 \), and \( \frac{p-1}{2.5} = 1 \); so that in this case, the only equation of exclusion is \( x+1 = 0 \), which admits only one solution, viz. \( x=p-1=10 \). Therefore all the non-residual numbers except 10 are the primitive roots; namely, 2, 6, 7, 8. We may extend this conclusion to every case when \( \frac{p-1}{2} \) is a prime number, as 7, 23, 47, &c.; in all which instances all the non-residuals, except \( p-1 \), are the primitive roots.

Next, let \( p=17 \); then \( \frac{p-1}{2} = 8 = 2^3 \); and there are no equations of exclusion. In this case, therefore, all the non-residuals, without exception, are primitive roots; and the same thing is true of every prime number of the form \( 2^n+1 \), such as 5, 257, &c.

Let \( p=13 \); then \( \frac{p-1}{2} = 2 \times 3 \); and the only equation of exclusion is \[ x^2 + 1 = 0, \] which admits only two solutions, viz. \( x=5 \) and \( x=8 \). In this instance, therefore, all the non-residual numbers, except 5 and 8, are the primitive roots.

Let \( p=31 \); then \( \frac{p-1}{2} = 3 \times 5 \); and we have two equations of exclusion, viz. \[ x^3 + 1 = 0 \\ x^5 + 1 = 0. \]

The non-residual numbers are 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30. Of these numbers the first, viz. 3, is a primitive root, since it satisfies neither of the two equations; and as the numbers less than 30, and prime to it, are 1, 7, 11, 13, 17, 19, 23, 29; all the primitive roots of 31 are as follows: viz. \( 3^1=3, 3^7=17, 3^{11}=13, 3^{15}=24, 3^{17}=22, 3^{19}=12, 3^{23}=11, 3^{29}=21 \). With respect to the other non-residual numbers, it will be found on trial, that the first equation is satisfied by 6 and 26; the second by 15, 23, 27, 29; and both equations by 30.

We are now prepared to enter upon the solution of the problem for dividing the circle into as many equal parts as there are units in the prime number \( p=2n+1 \). If we conceive a polygon of \( p \) sides, to be inscribed in a circle, it will be admitted that the centre of gravity of the polygon coincides with the centre of the circle. Therefore, if perpendiculars be drawn to any diameter of the circle from all the angles of the polygon, it follows, from the nature of the centre of gravity, that the sum of the cosines lying on one side of the centre of the circle will be equal to the sum of the cosines lying on the other side. Let \( \varphi = \frac{360^\circ}{p} \); and put \( u \) for the arc intercepted between the diameter and any angle of the polygon, then we shall have this equation, viz. \[ o = \cos u + \cos(\varphi+u) + \cos(2\varphi+u) + \ldots + \cos(2n\varphi+u), \] which is no more than the analytical expression of the geometrical property just mentioned. Now, suppose that the diameter passes through one of the angles of the polygon; then \( u=0 \), and the equation becomes \[ o = 1 + \cos \varphi + \cos 2\varphi + \cos 3\varphi + \ldots + \cos 2n\varphi. \] Let \( a \) be one of the primitive roots of the prime number \( p \); then rejecting multiples of \( p \), and paying no regard to the order, the terms of the geometrical progression, \[ a, a^2, a^3, a^4, \ldots, a^n, \] will be equal to the several numbers less than \( p \). Wherefore, in the two series of arcs, \[ a\varphi, a^2\varphi, a^3\varphi, a^4\varphi, \ldots, a^n\varphi, \\ \varphi, 2\varphi, 3\varphi, 4\varphi, \ldots, 2n\varphi, \] every arc in the geometrical progression will either be equal to some one in the arithmetical progression, or will differ from it by a whole circumference, or circumferences. Hence the cosines of the first series of arcs may be substituted in the last equation for the cosines of the other series; and thus we have \[ -1 = \cos a\varphi + \cos a^2\varphi + \cos a^3\varphi + \ldots + \cos a^n\varphi. \] Again, by Fermat's theorem, \( a^{2n}-1=(a^n+1) \). \( (a^n-1) \) is a multiple of \( p \); and because no primitive root of a prime number is the remainder of a square divided by that number, we have \( a^n+1 \) is a multiple of \( p \); and, consequently, \( a^{n+\lambda}+a^{\lambda} \) is a multiple of It follows, therefore, that \( a^{n+\lambda}\varphi + a^{\lambda}\varphi \) is equal to a multiple of the circumference of the circle; and hence,

\[ \cos a^{n+\lambda}\varphi = \cos a^{\lambda}\varphi \tag{A} \]

From this it appears that the cosines in the last equation may be distributed into two equal sums; one containing the cosines of all arcs from \( a\varphi \) to \( a^n\varphi \) inclusively, and the other the remaining cosines; consequently,

\[ -\frac{1}{2} = \cos a\varphi + \cos a^2\varphi + \cos a^3\varphi \ldots \cos a^n\varphi; \]

and because \( \cos a^n\varphi = \cos \varphi \),

\[ -\frac{1}{2} = \cos \varphi + \cos a\varphi + \cos a^2\varphi \ldots + \cos a^{n-1}\varphi. \tag{1} \]

Let \( \tau = \frac{360^\circ}{n} \); and put

\[ e = \cos \tau + \sin \tau \sqrt{-1}; \]

then all the powers of \( e \) with indices less than \( n \) will be different from one another, and all of them roots of the equation \( e^n - 1 = 0 \), the solution of which requires the division of the circle into only \( n \), or \( \frac{p-1}{2} \), equal parts.

In what follows, we shall have continual occasion to consider the expression

\[ \cos a^{\lambda}\varphi + e^m \cos a^{\lambda+1}\varphi + e^{2m} \cos a^{\lambda+2}\varphi \ldots + e^{(n-1)m} \cos a^{\lambda+n-1}\varphi; \]

and it will, therefore, be convenient to adopt an abridged mode of writing it. Now, the expression will be wholly known, and can be constructed when the two indices \( \lambda \) and \( m \) are given; and we may therefore denote it by the symbol \( f(\lambda, m) \), placing always the index of \( \lambda \) before the other. We shall invariably make the index of \( \lambda \) positive, and suppose it reduced below \( n \) by means of the formula (A). In like manner we shall suppose that the index of \( e \) is always reduced below \( n \) by suppressing the multiples of \( n \); and we shall write it sometimes positive and sometimes negative, observing that the negative indices may be always rendered positive by supplying the proper multiples of \( n \); thus, \( e^{-im} = e^{n-im} = e^{2n-im} = e^{3n-im} \), &c.

According to the notation just explained, we have

\[ f(o, m) = \cos \varphi + e^m \cos a\varphi + e^{2m} \cos a^2\varphi \ldots + e^{(n-1)m} \cos a^{n-1}\varphi, \] \[ f(o, -m) = \cos \varphi + e^{-m} \cos a\varphi + e^{-2m} \cos a^2\varphi \ldots + e^{-(n-1)m} \cos a^{n-1}\varphi. \]

And because \( e^0 = e^n = e^{-n} = 1 \), the symbols \( f(o, o) \), \( f(o, n) \), \( f(o, -n) \), will represent the series of cosines in the equation (1); so that we have

\[ -\frac{1}{2} = f(o, o) = f(o, n) = f(o, -n). \]

The following formula is no more than a corollary from the preceding notation, viz.

\[ e^{-\lambda m} \times f(o, m) = f(\lambda, m). \tag{B} \]

By means of the trigonometrical formula in common use, any powers and products of the cosines of the arc \( \varphi \) and its multiples may be reduced to a series of terms, containing the like cosines multiplied by given coefficients. Wherefore, because \( \cos p\varphi = 1 \), and likewise, because the cosines of all arcs greater than \( p\varphi \), \( 2p\varphi \), \( 3p\varphi \), &c. may be reduced to the cosines of arcs less than \( p\varphi \), it follows that every rational and integral function of \( \cos \varphi, \cos 2\varphi, \cos 3\varphi, \) &c. may be brought under this form of expression, viz.

\[ A + B \cos \varphi + C \cos 2\varphi + D \cos 3\varphi \ldots + N \cos 2n\varphi. \]

Now, if we suppose the function we are considering to be such, that it retains the same value when any of the multiple arcs \( 2\varphi, 3\varphi, \) &c. is substituted for \( \varphi \), the transformed expression will be possessed of the same property. But, if we actually substitute the arcs \( 2\varphi, 3\varphi, \) &c. for \( \varphi \) in the foregoing expression, it will become successively

\[ A + B \cos 2\varphi + C \cos 4\varphi + D \cos 6\varphi + \text{&c.} \] \[ A + B \cos 3\varphi + C \cos 6\varphi + D \cos 9\varphi + \text{&c.} \] \[ \text{&c.} \]

each line containing the same cosines, although in a different order, because the series of arcs is the same when whole circumferences, or the multiples of \( p\varphi \) are rejected; and all these expressions cannot have the same value unless \( B = C = D = \text{&c.} \); that is, unless the expression be of this form, viz.

\[ A + B (\cos \varphi + \cos 2\varphi + \cos 3\varphi \ldots + \cos 2n\varphi), \]

which, in consequence of what was before proved, is equal to \( A - B \). It is, therefore, demonstrated that every rational and integral function of \( \cos \varphi, \cos 2\varphi, \cos 3\varphi, \) &c., which remains unchanged when any of the multiple arcs \( 2\varphi, 3\varphi, \) &c. is substituted for \( \varphi \), has, for its value, an expression without cosines, and depending only upon the nature of the function.

If we introduce the arcs in geometrical, instead of those in arithmetical progression, it is obvious that the substitution of the multiple arcs \( 2\varphi, 3\varphi, \) &c., for \( \varphi \), is equivalent to the changing of \( \varphi \) into \( a\varphi, a^2\varphi, a^3\varphi, \) &c.; and hence any rational and integral function of the cosines of \( \varphi \) and its multiples, which remains invariable when \( \varphi \) is changed into \( a\varphi, a^2\varphi, a^3\varphi, \) &c. is a quantity independent of the cosines, or has its value expressed by a function from which the cosines are eliminated.

What has now been proved will enable us to appreciate the advantage arising from the introduction of the arcs in geometrical, in place of those in arithmetical progression, in which principally consists the improvement that this theory owes to M. Gauss. The solution of the problem turns upon finding those functions of \( \cos \varphi, \cos 2\varphi, \cos 3\varphi, \) &c., which have determinate values independent of the cosines; which functions, it has been proved, remain invariable when any of the multiple arcs \( 2\varphi, 3\varphi, \) &c. is substituted for \( \varphi \). Now, although the substitution of any multiple arc, in place of the arc itself, always reproduces the same series of cosines, yet the order is irregular, and varies with every different multiple arc; and this circumstance makes it difficult to investigate what change the substitution will effect in a given function. On the other hand, by introducing the arcs in geometrical progression, the same order is still preserved, whatever substitution be made; and, by this means, every facility possible is obtained for investigating the functions sought.

The following properties are deducible from what has been proved. First, if \( m, m', m'' \), &c. be any numbers, none of which is equal to zero, or a multiple of \( n \), and such that their sum is equal to \( n \), or to a multiple of \( n \); the product

\[ f(o, m) \times f(o, m') \times f(o, m'') \text{ &c.} \]

will be independent of the cosines of \( \varphi \) and its multiples, or will be an expression containing only the powers of \( e \) multiplied by numeral coefficients.

For by the formula (B) we have

\[ -e^{\lambda m} \times f(o, m) = f(\lambda, m) \] \[ e^{-\lambda m'} \times f(o, m') = f(\lambda, m') \] \[ e^{-\lambda m''} \times f(o, m'') = f(\lambda, m'') \] &c.

Therefore, by multiplying and observing that

\[ e^{-\lambda m} \times e^{-\lambda m'} \times e^{-\lambda m''} \times \text{&c.} = 1, \]

because \( \lambda \times (m + m' + m'' + \text{&c.}) \) is a multiple of \( n \), we get

\[ f(o, m) \times f(o, m') \times f(o, m'') \text{ &c.} = f(\lambda, m) \times f(\lambda, m') \times f(\lambda, m'') \text{ &c.} \]

which shows that the product in question is not altered when \( \varphi \) is changed into \( a^{\lambda} \varphi \). Consequently, according to what was before proved, the product is independent of the cosines.

It follows, as a corollary, that the product

\[ f(o, m) \times f(o, -m) \]

is independent of the cosines.

Next, if \( m, m', m'' \), &c. be any numbers, and \( s = m + m' + m'' \) &c.; and if neither \( s \) nor any of the numbers \( m, m', m'' \), &c. be a multiple of \( n \), we shall have

\[ f(o, m) \times f(o, m') \times f(o, m'') \text{ &c.} = M \times f(o, s), \]

the quantity \( M \) being independent of the cosines, and containing only the powers of \( e \) multiplied by numeral coefficients.

For, by the property already demonstrated, and its corollary, we have

\[ f(o, m) \times f(o, m') \times f(o, m'') \times f(o, -s) = A \] \[ f(o, s) \times f(o, -s) = A'; \]

\( A \) and \( A' \) being quantities independent of the cosines. Therefore, by exterminating \( f(o, -s) \), we get

\[ f(o, m) \times f(o, m') \times f(o, m'') \times \text{&c.} = \frac{A}{A'} f(o, s). \]

The foregoing properties are the foundations of the theory. But it is not enough to establish the principles by a general demonstration: it is also necessary to be able to compute the numerical values that occur in the application to particular problems. Therefore, supposing that \( m \) and \( m' \) are two numbers, and \( s = m + m' \), none of the three numbers \( s, m, m' \), being a multiple of \( n \), it is proposed to find the value of \( A \) in the equation.

\[ f(o, m) \times f(o, m') = A \times f(o, s). \]

For this purpose, set down the several terms of \( f(o, m') \) in their order; and below them write the terms of \( f(o, m) \), placing first any term, as \( e^{\lambda m} \cos a^{\lambda} \varphi \), and the rest in their order, in this manner,

\[ \cos \varphi + e^{m'} \cos a \varphi + e^{2m'} \cos a^2 \varphi \ldots + e^{(n-1)m'} \cos a^{n-1} \varphi \] \[ e^{\lambda m} \cos a^{\lambda} \varphi + e^{(\lambda+1)m} \cos a^{\lambda+1} \varphi + e^{(\lambda+2)m} \cos a^{\lambda+2} \varphi \ldots \] \[ + e^{(\lambda+n-1)m} \cos a^{\lambda+n-1} \varphi. \]

Now, let every term in the lower line be multiplied into that which stands above it; and, separating the factor \( e^{\lambda m} \), which is common to each product, let the symbol \( e^{\lambda m} \times \Psi(\lambda) \) represent the sum of all the products; then

\[ \Psi(\lambda) = \] \[ \cos \varphi \cos a^{\lambda} \varphi + e^{\lambda} \cos a \varphi \cos a^{\lambda+1} \varphi \ldots + e^{(n-1)\lambda} \cos a^{n-1} \varphi \cos a^{\lambda+n-1} \varphi. \]

If we repeat this operation, so as to make every term of the lower line stand first in succession, it is evident that, by this means, every term of \( f(o, m') \) will be multiplied by all the terms of \( f(o, m) \); so that the sum of all the results will be the product sought. We therefore obtain

\[ f(o, m) \times f(o, m') = \] \[ \Psi(o) + e^{m'} \Psi(1) + e^{2m'} \Psi(2) \ldots + e^{(n-1)m'} \Psi(n-1). \]

Let \( a^{\lambda+1} = w \), and \( a^{\lambda-1} = w' \); then because the product of the cosines of two arcs is equal to half the sum of the cosines of the sum and difference of the two arcs, we shall have

\[ \Psi(\lambda) = \] \[ \frac{1}{2} \left\{ \cos w \varphi + e^{\varphi} \cos a \cdot w \varphi + e^{2\varphi} \cos a^2 \cdot w \varphi + \text{&c.} \right\} \] \[ + \frac{1}{2} \left\{ \cos w' \varphi + e^{\varphi} \cos a \cdot w' \varphi + e^{2\varphi} \cos a^2 \cdot w' \varphi + \text{&c.} \right\}. \]

In the first place, when \( \lambda = o \), \( w = 2, w' = o \); therefore,

\[ \Psi(o) = \] \[ \frac{1}{2} \left\{ \cos 2 \varphi + e^{\varphi} \cos a \cdot 2 \varphi + e^{2\varphi} \cos a^2 \cdot 2 \varphi \ldots + e^{(n-1)\varphi} \cos a^{n-1} \cdot 2 \varphi \right\} \] \[ + \frac{1}{2} \left\{ 1 + e^{\varphi} + e^{2\varphi} + e^{3\varphi} \ldots + e^{(n-1)\varphi} \right\}. \]

But \( e^n - 1 = o \); and hence \( e^{ns} - 1 = o \); or

\[ o = (1 - e^s) \left\{ 1 + e^s + e^{2s} + \text{&c.} \ldots + e^{(n-1)s} \right\}; \]

and, according to the value assumed for \( e \), the equation \( 1 - e^s = o \) cannot take place when \( s \) is not a multiple of \( n \); wherefore

\[ o = 1 + e^s + e^{2s} + e^{3s} \ldots + e^{(n-1)s}. \]

Now, if we put \( a^2 = 2 \), we shall get

\[ \Psi(o) = \frac{1}{2} f(i, s) = \] Equations.

\[ \frac{1}{2} \left\{ \cos a^i p + e^i \cos a^{i+1} \varphi + e^{2i} \cos a^{i+2} \varphi + \text{&c.} \right\}. \]

Wherefore, on account of the formula (B), we finally get

\[ \Psi(o) = \frac{1}{2} e^{-is} \times f(o,s). \]

Next, when \( \lambda \) is not equal to zero, let \( h(\lambda) \) and \( h'(\lambda) \) denote the numbers derived from \( \lambda \) by means of the equations

\[ a^\lambda + 1 = a^{h(\lambda)} \\ a^\lambda - 1 = a^{h'(\lambda)} ; \]

then, by substituting \( a^{h(\lambda)} \) and \( a^{h'(\lambda)} \) for \( w \) and \( w' \), we shall get

\[ \Psi(\lambda) = \frac{1}{2} f \left( h(\lambda), s \right) + \frac{1}{2} f \left( h'(\lambda), s \right); \]

and on account of the formula (B),

\[ \Psi(\lambda) = \left\{ \frac{1}{2} e^{-(\lambda)s} + \frac{1}{2} e^{-h'(\lambda)s} \right\} \cdot f(o,s). \]

Now, collecting all the parts in the expression of \( f(o,m) \times f(o,m') \), we shall get these formulae, viz.

\[ f(o,m) \times f(o,m') = A \times f(o,s) \] \[ A = \frac{1}{2} e^{-is} + \frac{1}{2} e^{-h(1)s} + \frac{1}{2} e^{-m-h'(1)s} \] \[ + \frac{1}{2} e^{2m-h(2)s} + \frac{1}{2} e^{2m-h'(2)s} \] \[ + \frac{1}{2} e^{3m-h(3)s} + \frac{1}{2} e^{3m-h'(3)s} \] \[ \text{&c.} \qquad \text{&c.} \]

As nothing changes in the expression of A except the indices \( m \) and \( s \), it may be denoted by the abridged symbol \( (m,s) \), in which it is obvious that \( m' \) may be substituted for \( m \); so that

\[ A = (m,s) = (m',s). \]

When \( s \) is equal to \( n \), and \( m' = n - m \), the product in question becomes \( f(o,m) \times f(o,-m) \), which has been proved to be a quantity independent of the cosines. In this case, therefore, we shall have

\[ f(o,m) \times f(o,-m) = B; \]

\( B \) being a quantity from which the cosines are eliminated, and which is now to be investigated.

If, in the foregoing case, we suppose \( m' = n - m \) and \( s = n \), we shall get

\[ f(o,m) \times f(o,-m) = \Psi(o) + e^m \Psi(1) + e^{2m} \Psi(2) \ldots e^{(n-1)m} \Psi(n-1); \]

but here, because \( e^s = e^n = 1 \), \( e \) and its powers disappear from the expression of \( \Psi(\lambda) \), and we have

\[ \Psi(\lambda) = \cos \varphi \cos a^\lambda \varphi + \cos a \varphi \cos a^{\lambda+1} \varphi + \cos a^2 \varphi \times \cos a^{\lambda+2} \varphi + \text{&c.}; \]

and, by expanding the products of the cosines, as before,

\[ \Psi(\lambda) = \frac{1}{2} \left\{ \cos w \varphi + \cos a \cdot w \varphi + \cos a^2 w \varphi \ldots + \cos a^{n-1} \cdot w \varphi \right\} + \frac{1}{2} \left\{ \cos w' \varphi + \cos a \cdot w' \varphi + \cos a^2 \cdot w' \varphi \ldots + \cos a^{n-1} \cdot w' \varphi \right\}. \]

When \( \lambda = o, w = 2, w' = o \); therefore

\[ \Psi(o) = \frac{1}{2} \left\{ \cos 2 \varphi + \cos a \cdot 2 \varphi + \cos a^2 \cdot 2 \varphi \ldots + \cos a^{n-1} \cdot 2 \varphi \right\} + \frac{1}{2} \left\{ 1 + 1 + 1 + 1 \ldots \ldots + 1 \right\}. \]

But no alteration is made in equat. (1) when we substitute, instead of the arc \( \varphi \), any one of its multiples, or, which is the same thing, change \( \varphi \) into \( a \varphi, a^2 \varphi, \) &c.; because such substitution, or change, continually reproduces the same cosines. Thus it appears that the sum of the \( n \) cosines in \( \Psi(o) \), is equal to \( -\frac{1}{2} \); and we have

\[ \Psi(o) = \frac{n}{2} - \frac{1}{4}. \]

For every other value of \( \lambda, w \) and \( w' \) are, neither of them, equal to zero, nor to a multiple of \( n \); wherefore, according to what has just been said, the sum of the \( n \) cosines in each of the two parts of \( \Psi(\lambda) \), is equal to \( -\frac{1}{2} \); and thus, when \( \lambda \) is not equal to zero, we have

\[ \Psi(\lambda) = \frac{1}{2} \times -\frac{1}{2} + \frac{1}{2} \times -\frac{1}{2} = -\frac{1}{2}. \]

By substituting the values of \( \Psi(o) \) and \( \Psi(\lambda) \), we get

\[ f(o,m) \times f(o,-m) = \frac{n}{2} - \frac{1}{4} - \frac{1}{2} \left( e^m + e^{2m} + e^{3m} \ldots + e^{(n-1)m} \right). \]

But, as was already proved,

\[ -1 = e^m + e^{2m} + e^{3m} \ldots + e^{(n-1)m}; \]

wherefore,

\[ f(o,m) \times f(o,-m) = \frac{n}{2} - \frac{1}{4} + \frac{1}{2} = \frac{2n+1}{4} = \frac{1}{4} p. \]

Now, if we put \( k^2 = \frac{1}{4} p \), we have finally

\[ f(o,m) \times f(o,-m) = k^2 \ldots \ldots (3). \]

When \( n \) is an even number, it is obvious that

\[ f \left( o, \frac{n}{2} \right) = f \left( o, -\frac{n}{2} \right): \text{ therefore it follows as a corollary, that, in this case,} \] f \left( o, \frac{n}{2} \right) = f \left( o, -\frac{n}{2} \right) = \pm k = \pm \frac{1}{2} \sqrt{p}.

By applying the equat. (2) first to the indices m and m', and then to the indices n-m and n-m', or to m and m' taken negatively, we deduce

\[ f(o,m) \times f(o,m') = (m,s) \times f(o,s) \] \[ f(o,-m) \times f(o,-m') = (-m,-s) \times f(o,-s): \]

and, by multiplying, we shall get, on account of equat. (3), this remarkable formula, viz.

\[ (m,s) \times (-m,-s) = k^2 \ldots (4). \]

By successive applications of the equat. (2), we get

\[ f(o,1) \times f(o,1) = (1,2) \times f(o,2) \] \[ f(o,1) \times f(o,2) = (1,3) \times f(o,3) \] \[ f(o,1) \times f(o,3) = (1,4) \times f(o,4) \] &c.

By combining these equations, and writing P for \( f(o,1) \), we deduce

\[ P^2 = (1,2) \cdot f(o,2) \] \[ P^3 = (1,2) \cdot (1,3) \cdot f(o,3) \] \[ P^4 = (1,2) \cdot (1,3) \cdot (1,4) \cdot f(o,4) \] &c.

Wherefore, when n is an even number,

\[ P^n = (1,2) \cdot (1,3) \cdot (1,4) \ldots \left( 1, \frac{n}{2} \right) \cdot f \left( o, \frac{n}{2} \right); \]

and, by squaring and observing that, by equation (3),

\[ \left\{ f \left( o, \frac{n}{2} \right) \right\}^2 = k^2, \]

we get

\[ P^n = (1,2)^2 \cdot (1,3)^2 \cdot (1,4)^2 \ldots \left( 1, \frac{n}{2} \right)^2 \cdot k^2. \] (5).

When n is an odd number, we have in like manner,

\[ \frac{n-1}{2} \] \[ P^{\frac{n-1}{2}} = (1,2) \cdot (1,3) \cdot (1,4) \ldots \left( 1, \frac{n-1}{2} \right) \times f \left( o, \frac{n-1}{2} \right) \] \[ \frac{n+1}{2} \] \[ P^{\frac{n+1}{2}} = (1,2) \cdot (1,3) \cdot (1,4) \ldots \left( 1, \frac{n+1}{2} \right) \times f \left( o, \frac{n+1}{2} \right); \]

but, by equation (3), \( f \left( o, \frac{n-1}{2} \right) \times f \left( o, \frac{n+1}{2} \right) = k^2 \); wherefore

\[ P^n = (1,2)^2 \cdot (1,3)^2 \cdot (1,4)^2 \ldots \left( 1, \frac{n-1}{2} \right)^2 \cdot \left( 1, \frac{n+1}{2} \right)^2 \cdot k^2 \ldots (6). \]

Again, from the preceding expressions we get

\[ f(o,2) = \frac{1}{(1,2)} \cdot P^2; \]

and, by equation (4),

\[ f(o,2) = \frac{(-1,-2)}{k^2} \cdot P^2. \]

In like manner,

\[ f(o,3) = \frac{(-1,-2)}{k^2} \cdot \frac{(-1,-3)}{k^2} \cdot P^3 \] \[ f(o,4) = \frac{(-1,-2)}{k^2} \cdot \frac{(-1,-3)}{k^2} \cdot \frac{(-1,-4)}{k^2} \cdot P^4, \] &c.

These formulæ need only be continued till we obtain the value of the function \( \left( o, \frac{n-2}{2} \right) \) when n is even, and of \( f \left( o, \frac{n-1}{2} \right) \) when n is odd; the remaining functions \( f(o,n-2), f(o,n-3) \) &c. or, which is the same thing, \( f(o,-2), f(o,-3) \) &c. being derived from the preceding values merely by changing the signs of the different indices of e. Thus, if we write P' for \( f(o,-1) \), we shall have

\[ f(o,-2) = \frac{(1,2)}{k^2} \cdot P'^2 \] \[ f(o,-3) = \frac{(1,2)}{k^2} \cdot \frac{(1,3)}{k^2} \cdot P'^3 \] \[ f(o,-4) = \frac{(1,2)}{k^2} \cdot \frac{(1,3)}{k^2} \cdot \frac{(1,4)}{k^2} \cdot P'^4 \] &c.

Now, g being any number less than n, it has been shown that

\[ o = 1 + e^g + e^{2g} + e^{3g} \ldots + e^{(n-1)g}; \]

and hence if we attend to the nature of functions, \( f(o,o), f(o,1), f(o,2), \) &c. we shall readily get

\[ \cos a^g \varphi = \frac{f(o,o)}{n} + \frac{1}{n} \left\{ e^{-g} \cdot f(o,1) + e^{-2g} \cdot f(o,2) + e^{-3g} \cdot f(o,3) + \text{&c.} \right\}; \]

or, by arranging the terms differently, and because

\[ f(o,o) = -\frac{1}{2}, \]

\[ \cos a^g \varphi = -\frac{1}{2n} + \frac{1}{n} \left\{ e^{-g} \cdot f(o,1) + e^{g} \cdot f(o,-1) \right\} \] \[ + \frac{1}{n} \left\{ e^{-2g} \cdot f(o,2) + e^{2g} \cdot f(o,-2) \right\} \] \[ + \frac{1}{n} \left\{ e^{-3g} \cdot f(o,3) + e^{3g} \cdot f(o,-3) \right\} \] &c.

and it is to be observed that, when n is even, the last term is the single quantity \( \frac{1}{n} \times e^{-\frac{n}{2}g} \times f \left( o, \frac{n}{2} \right) \), which has no corresponding part. Now, this quantity is entirely known. For, since \( e^n = 1 \), we have Equations.

\[ e^{\frac{n}{2}} = e^{-\frac{n}{2}} = \pm 1; \text{ but } e \text{ has been so assumed, that none of its powers with indices less than } n \text{ are equal to unit; and, therefore, } e^{-\frac{n}{2}} = -1, \text{ and } e^{-\frac{n^2}{2}} = (-1)^{\frac{n}{2}}. \]

Again, by equation (3), \( f\left(o, \frac{n}{2}\right) = \pm k \); wherefore we have

\[ \frac{1}{n} \cdot e^{-\frac{n^2}{2}} \cdot f\left(o, \frac{n}{2}\right) = \frac{1}{n} \cdot (-1)^{\frac{n}{2}} \cdot \times \pm k. \]

On the whole, the preceding analysis brings us to the following formulae, which contain the solution of the problem, viz.

when \( n \) is even by equation (5),

\[ P^2 = (1, 2) \cdot (1, 3) \cdot (1, 4) \cdots \left(1, \frac{n}{2}\right) \times \pm k; \]

when \( n \) is odd, by equation (6),

\[ P^n = (1, 2)^2 \cdot (1, 3)^2 \cdot (1, 4)^2 \cdots \left(1, \frac{n-1}{2}\right)^2 \cdot \left(1, \frac{n+1}{2}\right) \cdot k^2; \]

and by equation (2), \( PP' = k^2 \).

Finally, by substituting the values of \( f(o, 2) \), \( f(o, 3) \), &c. \( f(o, -2), f(o, -3) \), &c. in the expression of \( \cos a^6 \varphi \), we get

\[ \cos a^6 \varphi = -\frac{1}{2n} + \frac{k}{n} \left\{ \frac{e^{-\xi P}}{k} + \frac{e^{\xi P'}}{k} \right\} \] \[ + \frac{k}{n} \left\{ \frac{(-1, -2)}{k} \left( \frac{e^{-\xi P}}{k} \right)^2 + \frac{(1, 2)}{k} \left( \frac{e^{\xi P'}}{k} \right)^2 \right\} \] \[ + \frac{k}{n} \left\{ \frac{(-1, -2)(-1, -3)}{k} \left( \frac{e^{-\xi P}}{k} \right)^5 + \frac{(1, 2)(1, 3)}{k} \left( \frac{e^{\xi P'}}{k} \right)^5 \right\} \] \[ + \text{&c.} \]

the series of terms must be continued till the last index of \( \frac{e^{-\xi P}}{k} \) and \( \frac{e^{\xi P'}}{k} \) is \( \frac{n-1}{2} \) when \( n \) is odd, and \( \frac{n-2}{2} \) when \( n \) is even; and, in this last case, the quantity \( \frac{1}{n} \times (-1)^{\frac{n}{2}} \times \pm k \), must be added, prefixing to \( k \) the same sign that is given to it in the value of \( P^2 \).

The solution of the problem is thus reduced to the computation of the functions (1, 2), (1, 3), &c. which requires no more than the substitution of 1 for \( m \), and of 2, 3, 4, &c. successively for \( s \), in the expression of A, equation (2). The half of these functions that have negative indices are deduced from the other half, merely by changing the signs of Equations. the several indices of \( e \), or by means of equation (4). All the cosines sought are found by substituting o, 1, 2, 3, &c. successively for \( s \). Although the function P is susceptible of \( n \) different values, represented by \( x, ex, e^2x, \) &c.; yet the same cosines are deduced from any one of these values. By this means all ambiguity is avoided with regard to the system of values that represent the cosines; but the numerical value that must be attached to each particular cosine remains quite indeterminate, because \( \varphi \) may equally stand for \( \frac{360^\circ}{p}, 2 \times \frac{360^\circ}{p}, 3 \times \frac{360^\circ}{p}, \) &c. The adaptation of the numerical quantities to the geometrical cosines must be made out by means of their relative magnitudes; the largest number answering to the greatest cosine. But when the value of one cosine is fixed, the rest are unambiguously determined by means of their indices.

In the formula for \( \cos a^6 \varphi \) all the terms in which two quantities are combined have real values, although their forms are imaginary. But it is not difficult to transform them into equivalent quantities without the imaginary sign.

It is manifest that the functions (1, 2) and (−1, −2) are of this form, viz.

\[ (1, 2) = A + Be + Ce^2 + De^3 \ldots + Ne^{n-1} \] \[ (-1, -2) = A + Be^{-1} + Ce^{-2} + De^{-3} \ldots + Ne^{-(n-1)}, \]

A, B, C, &c. denoting given coefficients. But, we have generally

\[ e^{\lambda} = \cos \lambda \tau + \sin \lambda \tau \sqrt{-1} \] \[ e^{-\lambda} = \cos \lambda \tau - \sin \lambda \tau \sqrt{-1}; \]

wherefore, by combining the two expressions of (1, 2) and (−1, −2), we shall readily get

\[ \frac{(1, 2) + (-1, -2)}{2} = A + B \cos \tau + C \cos 2\tau + \text{&c.} \] \[ \frac{(1, 2) - (-1, -2)}{2} = B \sin \tau + C \sin 2\tau + \text{&c.} \]

But, on account of equation (4), we may assume

\[ (1, 2) = k (\cos \beta + \sin \beta \sqrt{-1}) \] \[ (-1, -2) = k (\cos \beta - \sin \beta \sqrt{-1}); \]

and, by substituting these values in the last expressions, we get

\[ k \cos \beta = A + B \cos \tau + C \cos 2\tau + \text{&c.} \] \[ k \sin \beta = B \sin \tau + C \sin 2\tau + \text{&c.} \]

by which means the arc \( \beta \) is determined without ambiguity, since both its sine and cosine are ascertained. In like manner are determined the several arc in the formulae,

\[ (1, 3) = k (\cos \beta' + \sin \beta' \sqrt{-1}) \] \[ (-1, -3) = k (\cos \beta' - \sin \beta' \sqrt{-1}) \] \[ (1, 4) = k (\cos \beta'' + \sin \beta'' \sqrt{-1}) \] \[ (-1, -4) = k (\cos \beta'' - \sin \beta'' \sqrt{-1}) \] \[ \text{&c.} \] Again, because PP' = k², we may assume P = k (cos w + sin w √(-1)) P' = k (cos w - sin w √(-1)):

And if these values, and the similar values of the functions (1, 2), (1, 3), &c. be substituted in the value of

\[ \frac{n}{2} \cdot w = \beta + 3' + \beta'' + \text{&c.} \]

When n is an odd number, we must separate the function \( \left(1, \frac{n+1}{2}\right) \) from the rest, by supposing

\[ \left(1, \frac{n+1}{2}\right) = k (\cos \gamma + \sin \gamma \sqrt{-1}) \]

and then, by means of equation (6), we shall easily obtain

\[ nw = 2(\beta + \beta' + \beta'' + \text{&c.}) + \gamma. \]

The two last formulae determine the arc w; and we likewise have

\[ \frac{e^{-\epsilon P}}{k} = \cos(w - \frac{\pi}{3} r) + \sin(w - \frac{\pi}{3} r) \sqrt{-1} \] \[ \frac{e^{\epsilon P'}}{k} = \cos(w - \frac{\pi}{3} r) - \sin(w - \frac{\pi}{3} r) \sqrt{-1}; \]

and, by putting \( w^{(\epsilon)} = w - \frac{\pi}{3} r \),

\[ \frac{e^{-\epsilon P}}{k} = \cos w^{(\epsilon)} + \sin w^{(\epsilon)} \sqrt{-1} \] \[ \frac{e^{\epsilon P'}}{k} = \cos w^{(\epsilon)} - \sin w^{(\epsilon)} \sqrt{-1}. \]

Finally, by substituting the different values exhibited above in the formula for cos \( a^{\epsilon} \varphi \), we shall get

\[ \cos a^{\epsilon} \varphi = -\frac{1}{2n} + \frac{2k}{n} \cdot \cos w^{(\epsilon)} \] \[ + \frac{2k}{n} \cdot \cos(2w^{(\epsilon)} - \beta) \] \[ + \frac{2k}{n} \cdot \cos(3w^{(\epsilon)} - 3' - 3'') \] \[ + \frac{2k}{n} \cdot \cos(4w^{(\epsilon)} - \beta - \beta' - \beta'') \] \& c.

the series of terms being continued till all the arcs \( \beta, \beta', \beta'' \), &c. are taken in when n is odd; and till they are all taken in except the last when n is even,

in which case also the quantity \( (-1)^{\epsilon} \frac{k}{n} \) must be added.

By the preceding analysis the division of the circle into p equal parts is accomplished, when p is a prime number, by dividing a given arc into n or \( \frac{p-1}{2} \) equal parts. And this conclusion agrees with the general proposition of M. Gauss. For the nth part of a given arc is found by bisecting as often as n is divisible by 2, trisecting as often as it is divisible by 3, and so on. When n is a power of two, as in the case of the polygon of 17 sides, the solution is effected by repeated bisections, and thus comes under the elementary geometry. Supposing the division of the circle to be accomplished, we must further resolve the quadratic equation

\[ x + \frac{1}{x} = 2 \cos \frac{\lambda \times 360^\circ}{p}, \]

in order to find the roots of the binomial equation

\[ x^p - 1 = 0. \]

The following examples are subjoined for the sake of illustrating the method of calculation. And, in the first place, we may take the case of \( p = 11 \) equivalent to finding the roots of the equation \( x^{11} - 1 = 0 \), which was first solved by Vandermonde, and has been considered both by Lagrange and Legendre. Here,

\[ n = 5; k = \frac{1}{2} \sqrt{11}; r = \frac{360^\circ}{5} = 72^\circ; e = \cos r + \sin r \sqrt{-1}; \]

and, as 2 is a primitive root of 11, we may suppose \( a = 2 \). In order to find the numbers \( h(\lambda) \) and \( h'(\lambda) \), write down the series 1, 2, 3, &c., as far as n or 5; and, above each number, write the power of a equal to it when the multiples of 11 are rejected, taking always the least remainder, whether positive or negative: thus,

\[ \begin{array}{cccccc} a^0 & a^1 & a^3 & a^2 & a^4 \\ 1 & 2 & 3 & 4 & 5. \end{array} \]

In this arrangement of the powers of a, it is evident that, \( \lambda \) denoting any index, \( h(\lambda) \) is the next on the right hand, and \( h'(\lambda) \) the next on the left hand: we have, therefore,

\[ \begin{array}{ll} h(1) = 3 & h'(1) = 0 \\ h(2) = 4 & h'(2) = 3 \\ h(3) = 2 & h'(3) = 1 \\ h(4) = 4 & h'(4) = 2. \end{array} \]

Now, substitute these numbers in the expression of A, equat. (2), and likewise put \( m = 1 \); then,

\[ A = \frac{1}{2} e^{-s} + \frac{1}{2} e^{1-3s} + \frac{1}{2} e^{2-4s} + \frac{1}{2} e^{2-5s} + \frac{1}{2} e^{3-2s} + \frac{1}{2} e^{3-5s} + \frac{1}{2} e^{4-4s} + \frac{1}{2} e^{4-2s}. \]

In order to find (1, 2) and (1, 3) we have only to substitute 2 and 3 for s in the expression of A; hence

\[ (1, 2) = 1 + 2e + \frac{1}{2} e^3 + e^4; \] \[ (1, 3) = 1 + \frac{1}{2} e + 2e^2 + e^5; \]

which values will, in this case, be rendered some- Equations. what more simple by combining them with the equation \( o = 1 + e + e^2 + e^3 + e^4 \); and thus we get

\[ (1, 2) = -e^2 - \frac{1}{2}e^3 = m \] \[ (1, 3) = -e^2 - e^4 - \frac{1}{2}e^4 = \mu. \]

The functions \((-1, -2)\) and \((-1, -3)\) are found by subtracting the indices of \(e\) in the values of \((1, 2)\) and \((1, 3)\) from 5, which is equivalent to changing the signs of the indices: therefore, \[ (-1, -2) = -e^4 - e^3 - \frac{1}{2}e^2 = m \] \[ (-1, -3) = e^3 - e - \frac{1}{2}e^4 = \mu'. \] And it will be found, by actually multiplying, that \[ mm' = k^2 = \frac{11}{4} \quad \text{and} \quad \mu\mu' = k^2 = \frac{11}{4}. \] These values being found, we have, according to the foregoing method, \[ P^2 = (1, 2)^2 \cdot (1, 3) \cdot k^2 = m^2 \mu \cdot k^2 \] \[ P^{15} = \frac{k^{10}}{k^5} = (-1, -2)^2 \cdot (-1, -3) \cdot k^2 = m'^2 \mu' \cdot k^2 : \] and hence \[ \frac{P}{k} = \frac{1}{k}(m^2 \mu \cdot k^2)^{\frac{1}{3}} \] \[ \frac{P'}{k} = \frac{1}{k}(m'^2 \mu' \cdot k^2)^{\frac{1}{3}} \] \[ \frac{m'}{k} \cdot \frac{P^2}{k^2} = \frac{1}{k}(m'^2 \mu' \cdot k^2)^{\frac{1}{3}} \] \[ \frac{m}{k} \cdot \frac{P'^2}{k^2} = \frac{1}{k}(m^2 \mu \cdot k^2)^{\frac{1}{3}}; \] wherefore we have \[ \cos a^6 \varphi = -\frac{1}{10} + \frac{e^{-\varepsilon}}{5} \cdot (m^2 \mu \cdot k^2)^{\frac{1}{3}} + \frac{e^{\varepsilon}}{5} \cdot (m'^2 \mu' \cdot k^2)^{\frac{1}{3}} + \frac{e^{-2\varepsilon}}{5} \cdot (m'^2 \mu' \cdot k^2)^{\frac{1}{3}} + \frac{e^{2\varepsilon}}{5} \cdot (m^2 \mu \cdot k^2)^{\frac{1}{3}}. \] If, in this expression, we make \(e = 0\), and substitute the numerical values of \(k^2\), and of \(e\) and its powers, in the quantities under the radical sign, the result will coincide with the formula of Vandermonde, and with the calculation of Lagrange.

The expression just found being imaginary, if it be required to reduce it to a form fit for calculation, we must begin with substituting the values of \(e\) and its powers in \(m\) and \(\mu\): then \[ m = (\cos \tau - \cos 2\tau - \frac{1}{2} \cos 3\tau) + (\sin \tau - \sin 2\tau - \frac{1}{2} \sin 3\tau) \sqrt{-1} \] \[ \mu = (\cos 2\tau - \cos 4\tau - \frac{1}{2} \cos \tau) + (\sin 2\tau - \sin 4\tau - \frac{1}{2} \sin \tau) \sqrt{-1}. \] Now, \(\cos \tau = \cos 4\tau = -\frac{1}{4} + \frac{1}{2} \sqrt{5}\), and \(\cos 2\tau = \cos 3\tau = -\frac{1}{4} - \frac{1}{4} \sqrt{5}\); also \(\sin \tau = -\sin 4\tau\), and \(\sin 2\tau = -\sin 3\tau\): wherefore, \[ m = (\cos \tau - \frac{3}{2} \cos 2\tau) + (\sin \tau - \frac{1}{2} \sin 2\tau) \sqrt{-1} \] \[ \mu = (\cos 2\tau - \frac{3}{2} \cos \tau) + (\sin 2\tau + \frac{1}{2} \sin \tau) \sqrt{-1}. \] Again, \[ m = k (\cos \beta + \sin \beta \sqrt{-1}) \] \[ \mu = k (\cos \gamma + \sin \gamma \sqrt{-1}); \] consequently, \[ \cos \beta = \frac{1}{k} (\cos \tau - \frac{3}{2} \cos 2\tau) = \frac{1 + 5 \sqrt{5}}{4 \sqrt{11}}, \] \[ \sin \beta = \frac{1}{k} (\sin \tau - \frac{1}{2} \sin 2\tau) \] \[ \cos \gamma = \frac{1}{k} (\cos 2\tau - \frac{3}{2} \cos \tau) = \frac{-1 + 5 \sqrt{5}}{4 \sqrt{11}} \] \[ \sin \gamma = \frac{1}{k} (\sin 2\tau + \frac{1}{2} \sin \tau). \] Hence \[ \begin{align*} \beta &= 23^\circ 20' 46'' \\ \gamma &= 140^\circ 7' 6\frac{1}{2} \\ 5\omega &= 23 + \gamma = 186^\circ 48' 38\frac{1}{2} \\ \omega &= 37^\circ 21' 44'' \\ \omega' &= \omega - \varepsilon \times 72^\circ: \end{align*} \] \[ \cos a^6 \varphi = -\frac{1}{10} + \frac{\sqrt{11}}{5} \left\{ \cos \omega^{(e)} + \cos (2\omega^6 - \beta) \right\}. \] By making \(e\) successively equal to 0, 1, 2, 3, 4, the formula will give all the ten cosines of a polygon of 11 sides inscribed in a circle; because \(\cos \frac{360^\circ}{11}\) \[ = \cos 10 \cdot \frac{360}{11}, \cos 2 \cdot \frac{360}{11} = \cos 9 \cdot \frac{360}{11}, \text{etc.} \] It determines also the order of the arcs to which the numerical quantities belong; so that when the value of one cosine is fixed, the values of all the rest are likewise ascertained.

This last formula coincides with the calculation of Legendre.

The next example shall be the case of \(p = 17\). Then, \(n = 8, k = \frac{1}{2} \sqrt{17}, \tau = \frac{360}{8} = 45^\circ\), and \(e = \cos \tau + \sin \tau \sqrt{-1}\); and, 3 being one of the primitive roots of 17, we may take \(a = 3\). Now, arranging the powers of \(a\) as in the last example, we have \[ \begin{array}{ll} a^0, & a^6, \quad a^4, \quad a^5, \quad a^7, \quad a^5, \quad a^2 \\ 1, & 2, \quad 3, \quad 4, \quad 5, \quad 6, \quad 7, \quad 8: \end{array} \] and hence, \[ \begin{array}{ll} i = 6 & \\ h(1) = 4 & h'(1) = 6 \\ h(2) = 2 & h'(2) = 3 \\ h(3) = 2 & h'(3) = 7 \end{array} \] h(4)=5 h'(4)=1 h(5)=7 h'(5)=4 h(6)=1 h'(6)=0 h(7)=3 h'(7)=5.

By substituting these numbers in the expression of A, and likewise by putting m=1, we get

\[ A = \frac{1}{2} e^{-6s} + \frac{1}{2} e^{1-4s} + \frac{1}{2} e^{1-6s} + \frac{1}{2} e^{2-2s} + \frac{1}{2} e^{2-3s} + \frac{1}{2} e^{3-2s} + \frac{1}{2} e^{3-7s} + \frac{1}{2} e^{4-5s} + \frac{1}{2} e^{4-s} + \frac{1}{2} e^{5-7s} + \frac{1}{2} e^{5-4s} + \frac{1}{2} e^{6-4s} + \frac{1}{2} e^{6} + \frac{1}{2} e^{7-3s} + \frac{1}{2} e^{7-5s}. \]

In order to have the functions (1, 2), (1, 3), (1, 4), nothing more is necessary than to substitute 2, 3, 4 for s in the expression of A: then, observing that \(e+e^5=0\), \(e^2+e^8=0\), \(e^3+e^7=0\), we readily get

\[(1, 2) = \frac{3}{2} e^4 + e^7 + e^5 = \frac{3}{2} - \sqrt{-2} = -m\] \[(1, 3) = 1 + \frac{1}{2} e^4 + 2e^8 = \frac{1}{2} - 2\sqrt{-1} = n\] \[(1, 4) = \frac{3}{2} + e + e^3 = \frac{3}{2} + \sqrt{-2} = m:\]

and hence,

\[( -1, -2 ) = \frac{3}{2} e^4 + e + e^5 = \frac{3}{2} + \sqrt{-2} = -m'\] \[( -1, -3 ) = 1 + \frac{1}{2} e^4 + 2e^8 = \frac{1}{2} + 2\sqrt{-1} = n'\] \[( -1, -4 ) = \frac{3}{2} + e^7 + e^5 = \frac{3}{2} - \sqrt{-2} = m'\]

These values being found, we next have

\[ P' = (1, 2), (1, 3), (1, 4), f(0, 4); f(0, 4) = \pm k; \] therefore, making \(f(0, 4) = -k\), \[ P^4 = m^2 n k \] \[ P'^4 = \frac{k^8}{P^4} = m'^2 n' k : \] and hence, \[ \frac{1}{k} P = \frac{1}{k} (m^2 n k)^{\frac{1}{4}} ; \quad \frac{1}{k} \cdot P' = \frac{1}{k} (m'^2 n' k)^{\frac{1}{4}}; \] \[ -\frac{m'}{k} \cdot \frac{P^2}{k^2} = -\frac{1}{k} \cdot \sqrt{n k}; \quad -\frac{m}{k} \cdot \frac{P'^2}{k^2} = -\frac{1}{k} \cdot \sqrt{n' k}; \] \[ -\frac{m'}{k} \cdot \frac{n'}{k} \frac{P^5}{k^3} = -\frac{1}{k} (m^2 n' k)^{\frac{5}{4}}; \quad -\frac{m}{k} \cdot \frac{n}{k} \frac{P'^3}{k^3} = -\frac{1}{k} (m'^2 n k)^{\frac{3}{4}}; \] \[ \frac{1}{k} (m^2 n k)^{\frac{1}{4}}; \] \[ \cos a^6 \varphi = -\frac{1}{16} - (-1)^6 \cdot \frac{\sqrt{17}}{16} + \frac{1}{8} \left\{ e^{-\xi} (m^2 n k)^{\frac{1}{4}} + e^{\xi} (m'^2 n' k)^{\frac{1}{4}} \right\} - \frac{1}{8} \left\{ e^{-2\xi} (n k)^{\frac{1}{4}} + e^{2\xi} (n' k)^{\frac{1}{4}} \right\} - \frac{1}{8} \left\{ e^{-3\xi} (m^2 n' k)^{\frac{1}{4}} + e^{3\xi} (m'^2 n k)^{\frac{1}{4}} \right\}. \]

In order to reduce this expression, we shall put \[ \varphi(\xi) = e^{-2\xi} \sqrt{n k} + e^{2\xi} \sqrt{n' k} \] \[ \Psi(\xi) = e^{-\xi} (m^2 n k)^{\frac{1}{4}} + e^{\xi} (m'^2 n' k)^{\frac{1}{4}} - e^{-5\xi} (m^2 n' k)^{\frac{1}{4}} - e^{5\xi} (m'^2 n k)^{\frac{1}{4}}. \] And, because \(e^4 = e^{-4} = -1\), we get \(e^{6\xi} = e^{-2\xi} = (-1)^6 \cdot e^{2\xi}\), and \(e^{-6\xi} = e^{2\xi} = (-1)^6 \cdot e^{-2\xi}\). Therefore, by squaring, \[ \left\{ \varphi(\xi) \right\}^2 = 2k^2 + (-1)^6 \cdot k \] \[ \left\{ \Psi(\xi) \right\}^2 = 4k^2 - 6k(-1)^6 + 3\varphi(\xi) - 2k \cdot \varphi(\xi) \cdot (-1)^6. \] Now, in the formula for \(\cos a^6 \varphi\), viz. \(\cos a^6 \varphi = -\frac{1}{16} - (-1)^6 \cdot \frac{\sqrt{17}}{16} - \frac{1}{8} \varphi(\xi) - \frac{1}{8} \Psi(\xi)\), if we change \(\xi\) into \(\xi + 4\), no alteration will be produced, except that \(\Psi(\xi)\) will change its sign; for, it is obvious, that \[ \Psi(\xi + 4) = e^4 \cdot \Psi(\xi) = -\Psi(\xi). \] Hence, we readily deduce these two equations, viz. \[ \frac{1}{2} (\cos a^6 \varphi + \cos a^6 + \varphi) = \frac{1}{16} - (-1)^6 \cdot \frac{\sqrt{17}}{16} - \frac{1}{8} \varphi(\xi) \] \[ \frac{1}{4} (\cos a^6 \varphi - \cos a^6 + \varphi)^2 = \frac{1}{64} \times \left\{ \Psi(\xi) \right\}^2. \] If we suppose \(\xi = 0\), then \[ \left\{ \varphi(0) \right\}^2 = 2k^2 + k; \] wherefore, \[ \frac{1}{2} (\cos \varphi + \cos a^6 \varphi) = -\frac{1}{16} - \frac{\sqrt{17}}{16} - \frac{1}{8} \sqrt{2k^2 + k}, \] \[ \frac{1}{4} (\cos \varphi - \cos a^6 \varphi)^2 = \frac{1}{64} \cdot \left\{ 4k^2 - 6k - (2k - 3) \times \sqrt{2k^2 + k} \right\}. \] And, when \(\xi = 2\), then \[ \varphi(2) = e^4 \cdot \varphi(0) = -\varphi(0); \] wherefore, Equations.

\[ \frac{1}{2}(\cos a^2 \varphi + \cos a^6 \varphi) = -\frac{1}{16} - \frac{\sqrt{17}}{16} + \frac{1}{8} \sqrt{2k^2 + k} \]

\[ \frac{1}{4} (\cos a^2 - \cos a^6 \varphi)^2 = \frac{1}{64} \cdot \left\{ 4k^2 - 6k + (2k - 3) \right. \] \[ \left. \times \sqrt{2k^2 + k} \right\}. \]

Next, suppose \( \varepsilon = 1 \), then

\[ \left\{ \varphi(1) \right\}^2 = 2k^2 - k; \]

wherefore,

\[ \frac{1}{2}(\cos a \varphi + \cos a^5 \varphi) = -\frac{1}{16} + \frac{\sqrt{17}}{16} + \frac{1}{8} \sqrt{2k^2 - k} \]

\[ \frac{1}{4} (\cos a \varphi - \cos a^5 \varphi)^2 = \frac{1}{64} \cdot \left\{ 4k^2 + 6k + (2k + 3) \right. \] \[ \left. \sqrt{2k^2 - k} \right\}. \]

and, finally, making \( \varepsilon = 3 \), we get

\[ \varphi(3) = e^i \times \varphi(1) = -\varphi(1): \]

wherefore,

\[ \frac{1}{2}(\cos a^3 \varphi + \cos a^7 \varphi) = -\frac{1}{16} + \frac{\sqrt{17}}{16} - \frac{1}{8} \sqrt{2k^2 - k} \]

\[ \frac{1}{4} (\cos a^3 \varphi - \cos a^7 \varphi)^2 = \frac{1}{64} \cdot \left\{ 4k^2 + 6k - (2k + 3) \right. \] \[ \left. \sqrt{2k^2 - k} \right\}. \]

These formulæ enable us to find the numerical values of all the cosines sought; observing always that \( \varphi \) is indeterminate, and varies with the primitive root from which the solution is deduced. (c. c.) TABLE OF THE ARTICLES AND TREATISES CONTAINED IN THIS VOLUME.

EDINBURGH. EDINBURGHSHIRE. EDRISI, OR ALDRISI. EDUCATION. EDWARDS (BRYAN). EDWARDS (JONATHAN). EGYPT. ELBA. ELECTRICITY. ELLiptic Turning. ELLiptograph. EMBANKMENT. EMIGRATION. ENGLAND. ENTOMOLOGY. ENTRE-DUERO-E-Minho. EQUATIONS (Addendum, p. 669). ERNESTI (JOHN AUGUSTUS). ESSEXSHIRE. EUROPE. EXCHANGE. FAROE ISLANDS. FERGUSON (ADAM, LL.D.) FERMANAGH, COUNTY. FERMAT (PETER DE). FICHTE (JOHN THEOPHILUS). FIFESHIRE. FILANGIERI (GAETAN). FISHERIES. FLINTSHIRE. FLORIDA. FLUENTS, OR INTEGRALS. FLUIDS, ELEVATION OF. FONTANA (FELIX). FOOD. FORFARSHIRE. FORSTER (JOHN REINHOLD). FORSTER (JOHN GEORGE ADAM). FOURCROY (ANTONY FRANCIS DE). FOX (THE RIGHT HONOURABLE CHARLES JAMES). FRANCE. FRISI (PAUL). FUNDING SYSTEM. GALIANI (FERDINAND). GALVANISM. GALWAY, COUNTY. GARVE (CHRISTIAN). GAS-LIGHTS. GENOVESI (ANTHONY). GERMANY. GLAMORGANSHIRE. GLASGOW. GLOUCESTERSHIRE. GOVERNMENT. Granada, New. Great Britain. Greece. Guatemala, Goetimala, or Gualtimala. Guiana, or Guyana. Guyton de Morveau (Baron Louis Bernard). Haddingtonshire. Hampshire. Heligoland. Herculaneum. Herefordshire. Hertford College. Hertfordshire. Heyne (Christian Gottlob). Himalaya Mountains. Holland. (See Netherlands, Kingdom of). Holland, New. (See South Wales, New). Home (John). Horticulture. ERRATA.

Dissertation Second, p. 66, line 10, for "the quantities of matter are as the mean distances," read "the qualities of matter are as the orbits of the mean distances."

Page 316, col. 1, line 31, for "canal below," read "canal between." —— 318, col. 1, second line from bottom, for "the fluid will there," read "will therefore." ——— col. 2, line 40, for "case where," read "case when." ——— —— line 56, read \( \frac{1}{2} \)H. —— 319, col. 1, line 40, for "outside of the force," read "outside of the four."

DIRECTIONS FOR PLACING THE PLATES.

<table> <tr> <th>PLATE</th> <th>LXXIV.</th> <th>LXXV.</th> <th>LXXVI.</th> <th>LXXVII.</th> <th>LXXVIII.</th> <th>to face page</th> </tr> <tr> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>74</td> </tr> <tr> <td></td> <td>LXXIX.</td> <td></td> <td></td> <td></td> <td></td> <td>98</td> </tr> <tr> <td></td> <td>LXXX.</td> <td></td> <td></td> <td></td> <td></td> <td>322</td> </tr> <tr> <td></td> <td>LXXX.*</td> <td></td> <td></td> <td></td> <td></td> <td>444</td> </tr> <tr> <td></td> <td>LXXXI.</td> <td>LXXXII.</td> <td>LXXXIII.</td> <td>LXXXIV.</td> <td></td> <td>462</td> </tr> <tr> <th>MAP OF EUROPE,</th> <td></td> <td></td> <td></td> <td></td> <td></td> <td>200</td> </tr> </table>

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