Atomic Theory, a species of philosophy lately introduced into Chemistry, which deserves to be fully explained in this place.

It is well known to have been the doctrine of some of the most eminent of the Greek philosophers, that the ultimate elements of matter consisted of atoms or particles incapable of farther division or diminution. This doctrine was adopted by Sir Isaac Newton, and, indeed, has been almost universally embraced by modern philosophers. But it was in chemistry alone that it could be applied with any advantage. The object of chemists was to determine the component parts of bodies, and to ascertain the different elements out of which all substances are compound. Considerable progress was gradually made in this difficult investigation. Thus it was soon ascertained that the salts, which constitute a very numerous class of bodies, contain always at least two constituents, namely, an acid and a base. Thus, saltpetre is a compound of the acid called nitric acid, and the base called potash. Now, how small a portion soever of saltpetre you examine, whether a grain or the millionth part of a grain, you will always find it to contain both the ingredients of which saltpetre is composed. The same thing holds with respect to all the other salts, and, indeed, with respect to all compound bodies whatever. Now, this could not be the case, unless these compounds were formed by the union of the minutest possible particles of the constituents with each other; that is to say, unless it were the ultimate atoms of the elements which united together and constituted the compound. Accordingly, it has been admitted as an axiom in chemistry, that chemical union consists in the combination of the atoms of bodies with each other.

Chemistry originated from the absurd pursuits of the alchemists; and many years elapsed before it was able to shake off its connection with the chimerical notions respecting the philosopher's stone and the universal medicine. Dr Cullen was perhaps the first man that viewed the science as constituting a great and important branch of natural philosophy. His views were followed out by Black, Cavendish, Priestley, and a cloud of other eminent men, who have added so much lustre to the scientific pre-eminence of Great Britain. Margraaf, Bergman, and Scheele, were the first scientific chemists who appeared on the Continent. Bergman, in particular, was of the most essential service to the science. Educated in those branches of mechanical philosophy which had already made such progress, and accustomed to the rigid accuracy of mathematical reasoning, he introduced the same correct views, the same precise reasoning, the same generalization to which he had been already habituated. Lavoisier, who had the advantage of a similar education, adopted a similar conduct. His industry was indefatigable, and he contributed much more than any other person to introduce that precision in experimenting, and that accuracy of reasoning, which characterize the chemists of the present day.

The exertions of these eminent men led to the discovery of the art of analysis, such, though in a rude state, as it is practised at present. Bergman, Kirwan, and Wenzel, particularly distinguished themselves by the analysis of the salts. Of these three, Wenzel is by far the most accurate. The result of their experiments was, the demonstration that salts, and, indeed, all compound bodies, are universally composed of the same constituents united in the same proportions. Thus, water is always a compound of 1 part by weight of hydrogen, and $7\frac{1}{2}$ parts of oxygen; sulphate of copper always contains equal weights of sulphuric acid and black oxide of copper; and sulphate of barytes is always a compound of 100 parts by weight of sulphuric acid, and 194 of barytes.

J. B. Richter, who was for sometime Mining Secretary at Breslau, and afterwards Arcanist in the Porcelain manufactory at Berlin, where he died on the 4th of April 1807, employed himself, during the whole of his life, in endeavouring to introduce the use of mathematics into chemistry. His inaugural dissertation, printed in 1789, was entitled *De usu mathematicos in chymia*. In the year 1792, he published the first part of a work entitled *Anfangsgründe der Stochiometrie, oder Messkunst chymischer Elemente; Foundation of Stochiometry, or geometry of the chemical elements*. This work he continued to publish in successive parts, in 1793, 1794, 1795, 1802. Richter observed, that when two neutral salts, which mutually decompose each other, are mixed together, the two new salts which are formed still retain the same neutral state as the two original ones from which they were formed. Thus, sulphate of potash, and nitrate of barytes, are two neutral salts, which decompose each other when mutually mixed; sulphate of barytes, and nitrate of potash, being formed both in a neutral state. This circumstance enabled him to examine the accuracy of the results obtained by preceding experimenters, and he showed that the numbers assigned both by Bergman, Kirwan, and Wenzel, for the constituents of the salts, were inaccurate, as they were unable to stand the test of this double decomposition. He was induced, in consequence, to make a new set of experiments, in order to determine the constituents of the salts with more precision, and these experiments occupied him about ten years. He first analyzed the alkaline and earthy muriates and sulphates, then the nitrates, then the fluutes, then the carbonates, oxalates, succinates, tartrates, citrates, acetates, and some others; and lastly the phosphates. He placed in a table the quantity of each of the bases required to saturate 100 parts of muriatic acid, beginning with that base of which the smallest quantity was necessary for the purpose, and terminating with the base of which the greatest quantity was requisite. Similar tables were drawn up, representing the weight of the different bases requisite to saturate 100 parts of sulphuric acid, nitric acid, carbonic acid, and all the other acids contained in the different genera of salts which he had subjected to analysis. A comparison of these tables with each other enabled him to draw the two following remarkable consequences: First, The different bases follow exactly the same order in all the tables, and that order is as follows:

- Alumina. - Magnesia. - Ammonia. - Lime. - Soda. - Strontian. - Potash. - Barytes.

Second, The numbers in each table constitute a series, which have the same ratio to each other in all the tables. Suppose, for example, that in the table representing the muriates, the quantity of potash requisite to saturate 100 parts of muriatic acid were three times as great as the quantity of alumina requisite to produce the same effect, the same thing would hold in the sulphates, nitrates, and all the other genera of salts. Three times as much potash would be requisite to saturate 100 sulphuric, nitric, or any other acid, as would be required of alumina. In the same way, if three times as much barytes be requisite to saturate a given weight of one acid as is required of lime, the same proportion will hold when we saturate any other acid; with these earths we must always employ three times as much barytes as of lime. When the quantity of each of the acids requisite to saturate a given weight of each successive base, was placed in tables in a similar way, it was found, as might have been anticipated, that the acids followed precisely the same law.

These facts explain why, when two neutral salts decompose each other, the new formed salts are also neutral, and why there is never any excess of acid or base upon the one side or the other. The same proportions of bases that saturate a given weight of one acid, saturate all the other acids, and the same proportion of acids that saturate one base, saturate all the other bases. Hence numbers may be attached to each acid and base, indicating the weight of it which will saturate the weights attached to all the other bases or acids. This accordingly was done by Fischer from Richter's experiments and tables. We shall here insert his numbers, not because they are accurate, which is very far from being the case, but because they constitute the first specimen of the kind ever offered to the chemical world. (Berthollet's *Essai de Statique Chimique*, I. 136.)

| Alumina | Fluoric acid | |---------|-------------| | Magnesia | Carbonic acid | | Ammonia | Sebacic acid | | Lime | Muriatic acid | | Soda | Oxalic acid | | Strontian | Phosphoric acid | | Potash | Formic acid |

5 According to this table, 525 of alumina is saturated by 427 of fluoric acid, 377 of carbonic acid, 755 of oxalic acid, 1000 sulphuric acid, &c.; and 427 of fluoric acid, is saturated by 525 of alumina, 615 of magnesia, 672 of ammonia, &c. It is obvious, therefore, that this table, supposing it accurate, would give us the composition of all the salts into which the bases and acids contained in it enter.

M. Proust, a French chemist of great sagacity, who was settled as Professor of Chemistry at Madrid, was the first person who attempted an accurate examination of metallic oxides. He studied the oxides of iron, zinc, arsenic, antimony, &c. The result of his examination was, that every metal is capable of forming a certain determinate number of oxides and no more. Zinc forms only one oxide; iron, arsenic, and antimony, form two each; tin forms three, &c. Each of these oxides he found composed of a determinate proportion of metal, and oxygen, which remained always the same in all cases. Thus, the oxide of zinc, according to him, is always a compound of 100 parts zinc, and 25 oxygen. The two oxides of iron are the black and the red. If iron be combined with oxygen at all, it is either connected with the quantity of oxygen which constitutes the black oxide, or with the quantity which constitutes the red oxide, but it is incapable of uniting with any other proportion of oxygen. Wherever iron, already in the state of black oxide, combines with more oxygen, it becomes red oxide, being incapable of existing in any intermediate state between the black and the red oxide. The same law holds with respect to tin, antimony, arsenic, and all the metals; thus, M. Proust proved that metals unite only with determinate proportions of oxygen, which may be assigned in numbers. He showed that the same law held with respect to the combinations of the metals with sulphur. When Berthollet published his Chemical Statics, one of his objects was to show that there are two proportions of oxygen which unite with metals, a minimum and a maximum, and that between these two extremes, usually placed at a considerable interval from each other, there are an infinity of proportions of oxygen capable of uniting with every metal. Hence he inferred that there is no such thing as permanent definite metallic oxides; but an infinite number of shades of oxidization graduating into each other, and each capable of uniting with acids, and thus of forming an infinite variety of salts, differing from each other by minute shades of character. This extraordinary opinion, which, if correct, would have put an effectual period to all chemical investigation, as it would have been useless and absurd to attempt to examine compounds, which had no permanence and no analogy to each other, occasioned a discussion between Proust and Berthollet. Proust showed, by decisive experiments, that the indefinite numbers of metallic oxide, supposed by Berthollet, does not exist, and that metals are only capable of combining with 1, 2, or 3 doses of oxygen; that there is no indefinite number of oxides between the black and the red oxides of iron; but that the instant that iron combined with more oxygen than existed in the black oxide, it is converted into the red oxide.

Such was the state of the subject when Mr. Dalton turned his attention to the combination of bodies with each other about the year 1804. At that period, it was known that hydrogen combines only in one proportion with oxygen; that carbon, sulphur, and phosphorus, unite each in two proportions; that carbon unites in two proportions with hydrogen, sulphur in two proportions with the same substance, and phosphorus in one proportion; that azote unites in one proportion with hydrogen, and in four proportions with oxygen. It was known, too, that several of the metals combine with only one dose of oxygen, while some unite with two doses, and others with three. It had been ascertained, that, in certain cases, when metals combine with two doses of oxygen, the second dose is exactly double the first dose, while in other cases it is to the first dose as 3 to 2. Thus, in platinum, the second oxide contains just double the quantity of oxygen that the first oxide contains, while in iron, the quantity of oxygen in the red oxide, is, to the quantity in the black oxide, as 3 to 2. None of the combinations of oxygen or hydrogen, with the simple combustibles, had been accurately analyzed; though approximations to a correct analysis of most of them existed, and what was chiefly wanting, was a criterion for distinguishing the comparative merits of the different analysis of these bodies which had been presented to the chemical world.

Mr. Dalton was struck with the small number of compounds which the elementary bodies are capable of forming with each other, and with the very simple relations which existed between the numbers that denote the weight of the different constituents in these compounds; and he set himself down to endeavour to account for these circumstances not hitherto inquired into. Water, he found, was always a compound of 1 part by weight of hydrogen, and 8 parts by weight of oxygen.

| Compound | Weight | |---------------------------|--------| | Carbonic oxide | 750 | | Carbonic acid | 750 | | Sulphurous acid | 2 | | Sulphuric acid | 2 | | Nitrous oxide | 175 | | Nitrous gas | 175 | | Nitrous acid | 175 | | Nitric acid | 175 |

* In the following examples, instead of using the imperfect analysis with which Mr. Dalton was obliged to be satisfied, and which misled him in some cases, we have thought it better to employ the most precise experiments hitherto made. Now, in all these compounds, it will be perceived, that the numbers denoting the oxygen, united to a given weight of the same base, bear a very simple and obvious ratio to each other. The oxygen in carbonic acid is twice as great as in carbonic oxide; the oxygen in sulphuric acid is to that in sulphurous as the numbers 3 to 2; while the oxygen, in the respective compounds of azote and oxygen, bears the ratio of the numbers 1, 2, 3, 5.

These and many other similar examples which Mr Dalton registered and examined, but which it is unnecessary to state here, led him to the lucky idea that the atoms of bodies unite together; that the atom of each body has a determinate weight, and that this weight regulates the proportion in which bodies combine with each other. Let us suppose, for example, that water is formed by the union of one atom of oxygen, with one atom of hydrogen, it follows, as the oxygen in water is 8 times that of the hydrogen, that the weight of an atom of oxygen is to that of an atom of hydrogen as 8 to 1. So that, if we represent the weight of an atom of hydrogen by 1, that of an atom of oxygen will be 8; or if, as is more convenient in practice, we pitch upon 1 for the weight of an atom of oxygen, an atom of hydrogen will weigh 0.125. As carbon unites with two proportions of oxygen, we may suppose that an atom of carbon unites with 1 atom, and with 2 atoms of oxygen. Hence, the reason why the quantity of oxygen in carbonic acid is twice as great as in carbonic oxide. If an atom of oxygen weigh 1, it is obvious that an atom of carbon will weigh 0.751, supposing the numbers which we have given in the preceding table to be accurate. It is equally obvious, that sulphurous acid must contain 2 atoms of oxygen, and sulphuric acid 3 atoms; and that an atom of sulphur will weigh 2, on the supposition that an atom of oxygen weighs 1. Finally, nitrous oxide is a compound of 1 atom azote, and 1 atom oxygen; nitrous gas of 1 atom azote, and 2 atoms oxygen; nitrous acid of 1 atom azote, and 3 atoms oxygen; and nitric acid of 1 atom azote, and 5 atoms oxygen. And, supposing the numbers in our table correct, if an atom of oxygen weigh 1, an atom of azote will weigh 1.75.

Mr Dalton did not rest satisfied with this simple and luminous explanation, which threw a new and strong light around chemical combinations—which afforded the means of correcting and checking chemical experiments, hitherto conducted without any guide, and promised, in time, to introduce mathematical precision, and mathematical reasoning, into a science which hitherto has been able only to boast of analogical and probable conclusions. He contrived a set of symbols to represent the different elements, and make the nature of the combinations which they form obvious to the eye of the most careless reader. A few examples will be sufficient to make the nature and use of these symbols perfectly well understood.

Let \( O \) be an atom of oxygen

- \( O \) - - - - hydrogen - \( O \) - - - - azote - \( O \) - - - - carbon

Let \( \oplus \) be an atom of sulphur

- \( \oplus \) - - - - phosphorus,

Then \( \bigcirc \) represents an integrant particle

- \( \bigcirc \) - - - - of water - \( \bigcirc \) - - - - of carbonic oxide - \( \bigcirc \) - - - - of carbonic acid - \( \bigcirc \) - - - - of sulphurous acid - \( \bigcirc \) - - - - of nitrous oxide - \( \bigcirc \) - - - - of nitrous gas - \( \bigcirc \) - - - - of nitrous acid - \( \bigcirc \) - - - - of nitric acid - \( \bigcirc \) - - - - of carbureted hydrogen - \( \bigcirc \) - - - - of olefiant gas.

It would be easy to multiply these symbols much farther; but the preceding specimen is sufficient, we conceive, to make the use of them understood, and even to make Mr Dalton's doctrine more simple to those who are still strangers to it. Mr Dalton has already published two volumes of what he entitles a New System of Chemical Philosophy, in order to explain and show the accuracy and importance of this doctrine. Our object, in the present article, will be to lay the whole of it in its present state before our readers. But we must, in the first place, trace the history of the improvements which it has received to its termination.

In the year 1809, M. Gay-Lussac, who had already distinguished himself by many important chemical discoveries, and who was well acquainted with Mr Dalton's theory, published a curious paper on the combinations of gaseous substances with each other. (Memoires d'Arcueil, II. 205.) He showed that they follow a very simple law in their combinations; that one volume of one gas always combines either with one volume of another gas, or with two volumes, or with three volumes. Thus,

1 volume ammonia saturates 1 vol. of muriatic acid 1 vol. of carbonic acid 1 vol. of fluoboric acid.

The resulting compounds are neutral muriate, carbonate, and fluoborate of ammonia.

Nitrous gas is composed of one volume of azote, and one volume of oxygen.

Subcarbonate of ammonia is composed of one volume carbonic acid, and two volumes ammoniacal gas.

Subfluoborate of ammonia of one volume fluoboric acid, and two volumes ammoniacal gas. Water of one volume of oxygen, and two volumes of hydrogen gas. Nitrous oxide of one volume oxygen, and two volumes azote. Nitric acid of one volume oxygen, and two volumes nitrous gas. Sulphuric acid of one volume oxygen, and two volumes sulphurous acid. Carbonic acid gas of one volume oxygen, and two volumes carbonic oxide.

Finally, ammonia is composed of one volume azote, and three volumes hydrogen; and nitrous acid gas of one volume oxygen, and three volumes of nitrous gas.

Gay-Lussac showed likewise, that the diminution of volume which takes place on the combination of these gases, follows as simple a law as the volumes in which they combine. In nitrous gas, the oxygen and azote undergo no condensation. Accordingly, the specific gravity of nitrous gas is the mean between that of oxygen and azote. Frequently one of the two gases retains its bulk unaltered, while the other totally disappears. Thus, when 100 measures of carbonic oxide, and 50 measures of oxygen gas, are united together by combination, the resulting compound constitutes 100 measures of carbonic acid. Sometimes the bulk is reduced to one half. Thus, when 100 measures of azote, and 300 measures of hydrogen, are united, they form 200 measures of ammoniacal gas.

This curious law, discovered by Gay-Lussac, when coupled with the doctrine of Mr Dalton, shows us, that there exists a very simple relation between the weight of the atoms and the bulk of the gaseous bodies. Accordingly, the weight of the atoms of these bodies may be deduced, with sufficient accuracy, from their specific gravity. This method was employed both by Sir Humphry Davy and Dr Wollaston.

Professor Berzelius of Stockholm, one of the most accurate and indefatigable chemists of the present time, was led, some years ago, to study the composition of the salts, in consequence of the perusal of Richter's Stoichiometry. He soon satisfied himself that neither the analyses of Richter, nor of preceding chemists, could be depended on for accuracy. He resolved, therefore, to institute a new analysis of a considerable number of saline bodies, and to be at particular pains to ensure the greatest possible precision which the present state of the science enabled him to attain. The result of this laborious undertaking has been published in the Annales de Chimie, the Annalen der Physik, Nicholson's Journal, and the Annals of Philosophy. It constitutes by far the most complete and valuable set of chemical analyses of which the science is possessed. Berzelius not only satisfied himself of the accuracy of Richter's law, but was so fortunate as to discover another of a very unexpected nature, and of the greatest importance in chemical analysis, because it enables us to subject the results of all our analyses to calculation, and thus to determine with certainty how far their accuracy may be depended on. All the acids, with a very few exceptions, contain oxygen. This is the case likewise with all the salifiable bases, so that the salts may be considered as combinations of two sets of oxides with each other. Now, Berzelius compared the oxygen in the base of a salt with the oxygen in its acid, and he found that they always bore a very simple relation to each other. They were either equal, or the oxygen in the acid was twice as much as that in the base, or thrice as much, or four times as much, &c. If the acid contained twice as much oxygen as the base, this was a proof that the acid contained two atoms of oxygen; if it contained thrice as much, it contained three atoms of oxygen, and so on. Thus, in the sulphates, the sulphuric acid contains three times as much oxygen as the quantity of base which it saturates. Hence we infer, that sulphuric acid is a compound of 1 atom sulphur, and 3 atoms oxygen. In the sulphites, the acid contains twice as much oxygen as the base which it saturates. Hence, we infer that sulphurous acid is composed of 1 atom sulphur, and 2 atoms oxygen. In the carbonates, the oxygen in the acid is twice as much as in the base which it saturates. Hence we infer that carbonic acid is composed of 1 atom of carbon, and 2 atoms of oxygen.

Berzelius, during his experimental investigation of the atomic theory, met with some objections which he was not able immediately to surmount. In consequence of these difficulties, he substituted for the atomic theory what he calls the theory of volumes, derived from Gay-Lussac's discovery respecting the proportions in point of volume in which gaseous substances combine. This is merely substituting another name for the atomic theory. For it is easy to show, that the theory of volumes, as Berzelius considers it, is precisely the same thing as Dalton's atomic theory. The difficulties of Berzelius will come under our consideration hereafter.

After the preceding historical sketch, which we thought necessary in order to do justice to all parties concerned in this important branch of chemistry, we shall proceed to lay the atomic theory, with the proofs in support of it, before our readers.

I. Outline of the Atomic Theory.

With respect to the nature of the ultimate elements of bodies, we have no direct means of obtaining accurate information. But it is the general opinion, that they consist of atoms or minute particles, incapable of farther division. It is impossible to demonstrate the truth of this opinion; but, for our part, we can conceive no other. It has been almost universally admitted, by mechanical philosophers, since the time of Newton. We shall, therefore, adopt it here as the foundation of our reasoning.

When two bodies unite chemically, so as to form a third body, the two substances combined are dispersed everywhere through the new compound. Water, for example, is composed of oxygen and hydrogen. Now, how minute a portion soever of water we examine, we shall find it to contain both oxygen and hydrogen. Saltpetre is a compound of nitric acid and potash. If we examine the salt, whether we take an ounce or the hundredth part of a grain, we shall always find it to be a compound of nitric acid and potash. If any portion of it were to want one of these constituents, it would no longer be saltpetre; it would be potash, or nitric acid, according to the constituent which is not present. Limestone is a compound of carbonic acid and lime. Now, we may reduce it to an impalpable powder; but if we take one of the grains of this powder, however small, and throw it into nitric acid, we shall perceive an effervescence, indicating the presence of carbonic acid, and it will dissolve, indicating that lime was also one of its constituents. Mechanical trituration, however carefully made, is quite incapable of separating from each other substances which are chemically combined. Now, this distribution of the constituents could never be so complete through every part of the compound, unless it were the atoms of the combined bodies that united with each other. This, accordingly, is the opinion respecting chemical combinations, which appears always to have been entertained by chemical philosophers. Nor do we believe that any person will be disposed to call its truth in question.

All chemical compounds contain the same constant proportion of constituents with the most rigid accuracy, no variation whatever ever taking place. Water is universally a compound of 1 part by weight of hydrogen, and 8 of oxygen; sulphuric acid, of 1 part sulphur, and 1\(\frac{1}{2}\) oxygen; carbonic acid, of 1 part carbon, and 2.666 of oxygen. This permanency in the constituents of chemical compounds, indeed, is generally admitted. It constitutes, in fact, the basis of the whole science. For, if the proportions of the constituents of bodies were variable, chemical analysis would be nugatory, and the whole science of little importance. Even Berthollet, who contends for indefinite proportions in the abstract, admits the incontrovertible fact that the proportions of chemical combinations in general are permanent.

Since, then, bodies unite chemically atom to atom; and since they always unite in the same proportions, without the smallest deviation, this regularity can be ascribed to nothing else than the constant union of one atom of one body with one atom of another; or of a determinate number of atoms of one body with a determinate number of atoms of another. At first sight, it may appear impossible to determine how many atoms of each constituent combine with an indefinite number of atoms of the other. But there are some circumstances which afford us considerable assistance in this apparently intricate investigation.

Thus, oxygen has the property of uniting in different proportions with the same base; sometimes in two, sometimes in three, sometimes in four, &c., proportions. Thus, with carbon it unites in two proportions, with mercury in two, with copper in two. With tin it unites in three proportions, while with manganese it unites in four proportions. Now, if we represent the weight of base with which the oxygen unites by \(a\), and suppose all the different proportions of oxygen to unite with this proportion of base; and if we denote the first portion of oxygen by \(b\), then, in general, the constituents of the different compounds formed by the union of the different doses of oxygen with the base will be as follows:

1st compound \(a + b\). 2d compound \(a + 2b\). 3d compound \(a + 3b\). 4th compound \(a + 4b\).

Suppose 10 parts of oxygen enter into the first compound, then 20 parts enter into the second, 30 parts into the third, and 40 parts into the fourth compound. Hence, whatever number of atoms enter into the first compound, twice that number enters into the second compound, thrice the number into the third compound, and four times the number into the fourth compound. Hence it is clear, that there is a determinate number of atoms of oxygen, which always enter into these combinations. If we represent this number by \(x\), then \(a + x\) is the first compound, \(a + 2x\) the second, \(a + 3x\) the third, and \(a + 4x\) the fourth. Now, it would be singular, if 2, 3, 4, &c., atoms of oxygen, were to be always inseparably linked together, so as never to be able to enter into combinations separate. It is much more simple to conceive, that \(x\) represents only 1 atom. Indeed, there can be little doubt that this is actually the case. For oxygen gas, being a permanently elastic fluid, must consist of atoms that repel each other. Hence a compound atom of oxygen, or a number of atoms united together, seems to be impossible. But even though the opinion that \(x\) represents one atom should not be mathematically true, still it would be proper to adopt it. For, as far as our calculations are concerned, a number of atoms of oxygen, constantly and invariably united together, would constitute a compound atom, about which we may reason as accurately and justly as we could do about the simple atoms themselves. But if \(x\) in these combinations be considered as representing an atom of oxygen, there can be no doubt that \(a\) represents an atom of the body, which unites with oxygen. So that, by knowing the proportions in which they unite, we have the relative weight of an atom of oxygen, and of the body with which it unites.

This reasoning may be applied to hydrogen as well as oxygen. Hydrogen has the property of uniting in different proportions with various bodies, as with carbon, phosphorus, sulphur, &c. In these different proportions, we find the hydrogen always denoted by \(y\), \(2y\), &c., just as is the case with oxygen. Hence we have the same reason for concluding that \(y\), which represents the minimum of hydrogen which unites with the other constituent, is an atom.

The numbers \(x\) and \(y\) are easily discovered by making an accurate analysis of the different compounds into which various proportions of oxygen and hydrogen enter; and, when reduced to their lowest terms, they are very nearly \(x = 8\), and \(y = 1\). Hence these numbers represent the ratios of the weight of an atom of oxygen and an atom of hydrogen to each other. Now, it deserves attention, that these numbers represent the composition of water. For it has been ascertained, by very careful experiments, that water is composed of 100 measures of oxygen gas, and 200 measures of hydrogen gas. Now, the specific gravity of these gases is as follows:

| Substance | Specific Gravity | |-----------|-----------------| | Oxygen | 1.111 | | Hydrogen | 0.069 |

Hence water is composed by weight of

| Substance | Weight Ratio | |-----------|-------------| | Oxygen | 8.00 | | Hydrogen | 1.00 | From this coincidence we are entitled to conclude, that water is formed by the union of an atom of oxygen with an atom of hydrogen. And consequently, that an atom of hydrogen is to an atom of oxygen in weight, as 1 to 8. This very important conclusion is supported by other considerations. Oxygen and hydrogen have never been made to combine in any other proportion than that in which they exist in water. Hence this proportion must be that which unites most readily and with the greatest force. Now, as the atoms of hydrogen repel each other, as is the case also with the atoms of oxygen, it is obvious, that when they are mixed equably, as is the case when 200 measures of hydrogen gas, and 100 measures of oxygen gas, are put into a tube and fired by electricity, they will most readily unite atom to atom. This, though not in itself decisive, is a corroborating circumstance. It follows from it, that a given bulk of hydrogen gas contains only one half the number of atoms which exist in the same bulk of oxygen gas.

But we must not conceal that there is another view of this subject, which has been taken by Davy and Berzelius, and certainly possesses much plausibility. It is founded on this curious fact, first noticed, we believe, by Dr Wollaston, and confirmed by the theory of Gay-Lussac, detailed in the historical introduction to this article, that if we determine the weight of an atom of the gaseous bodies from their specific gravity (or at least a multiple of it), the numbers obtained will correspond with the atomic theory, as well as the numbers found by the method above described, which is the method of Dalton. In fact, they will always be the same numbers, or their halves or their doubles. This shows us, that a very simple relation exists between the bulk of a gas and the number of atoms which it contains. If the weight of an atom of a gas were always the same as its specific gravity, it would follow, that the same bulk of every gas contains always the same number of atoms. But cases occur in which this simple rule does not answer. Thus, it is admitted even by Davy and Berzelius, that the weight of an atom of azote is twice that of its specific gravity. So that, in a given measure of azotic gas, there is only one half of the number of atoms that exist in an equal measure of oxygen gas.

Those who found their weights of the atoms of bodies on the specific gravity of the gases, consider water as composed of 2 atoms of hydrogen, and 1 atom of oxygen. If this notion be correct, the weight of an atom of hydrogen, is to that of an atom of oxygen, as 1 to 16. This is nearly the proportion given by Davy and Berzelius. Davy, who rejects decimals, makes an atom of hydrogen weigh 1, and an atom of oxygen 15. Berzelius makes an atom of oxygen weigh 15.069, supposing that of hydrogen to weigh 1. As oxygen and hydrogen cannot be united in any other than 1 proportion, we have no means of putting it to the test of experiment. But the example of azote (admitted at least by Davy, though Berzelius, by his hypothesis, that azote is a compound of nitricum and oxygen, gets over the difficulty) is sufficient to show us that it is not necessary that the weights of the atoms of gases should be directly as the specific gravity of these bodies. On this account, the reasons before urged in favour of the opinion, that water is a compound of 1 atom of oxygen, and 1 atom of hydrogen, seems to us sufficient to give a preponderancy to this opinion, and to induce us to embrace it.

Knowing the weight of an atom of oxygen, and an atom of hydrogen, we have it in our power to determine the weight of an atom of the other substances, which unite with oxygen or with hydrogen, or with both. For example, 100 parts of sulphur unite with two proportions of oxygen, namely, with 100 parts constituting sulphurous acid, and with 150 parts constituting sulphuric acid, both by weight. Here, the proportions of oxygen being to each other as the numbers 1:1½ or 2:3; it is reasonable to suppose, that the first proportion represents 2 atoms of oxygen, and the second 3 atoms. There may, perhaps, exist another compound, consisting of 1 atom sulphur, and 1 atom oxygen, or by weight of 100 sulphur and 50 oxygen. But such a compound is at present unknown. It is evident, if sulphurous acid be composed of 1 atom sulphur, and 2 atoms oxygen, that an atom of sulphur is twice as heavy as an atom of oxygen. So that if we were to represent the weight of an atom of oxygen by 8, that of an atom of sulphur would be 16. Davy makes an atom of oxygen 15. Accordingly, his atom of sulphur is represented by 30.

We have it in our power to verify this reasoning by means of the combinations which sulphur makes with hydrogen. This verification may be made two ways, and we shall employ both to show how they serve to corroborate one another. 1. It has been ascertained, that 100 measures of hydrogen gas may be united to sulphur, and converted into sulphureted hydrogen gas, without undergoing any change of bulk. Hence, to know with precision the composition of sulphureted hydrogen, we have only to determine with care the specific gravity of hydrogen gas, and sulphureted hydrogen gas. Now,

100 cubic inches of hydrogen gas weigh 2.117 grains. 100 cubic inches of sulphur. hyd. gas 35.89.

Hence it follows, that sulphureted hydrogen gas is composed of

| Hydrogen | - - - | 2.117 or 1.00 | | Sulphur | - - - | 33.773 16.00 |

This shows us, that if sulphureted hydrogen gas is composed of 1 atom sulphur, and 1 atom hydrogen, and if we suppose an atom of hydrogen to weigh 1, an atom of sulphur will weigh 16. The combination of oxygen and sulphur gave us 16. Thus the two processes of reasoning lead to the same conclusion, since the difference really existing is greatly within the limits of the unavoidable errors in such kinds of experiment. 2. Sulphureted hydrogen gas requires for its complete combustion 1½ times its bulk of oxygen gas. The products of the combustions are sulphurous acid and water. The sulphurous acid gas amounts exactly to the bulk of the sulphureted hydrogen gas consumed. Suppose 100 measures of sulphureted hydrogen gas to be burned. We must employ for the purpose 130 measures of oxygen gas. The gaseous product consists of 100 measures of sulphurous acid gas; 100 measures of the oxygen gas go to the formation of the sulphurous acid gas, and 50 measures go to the formation of water. Now, 100 cubic inches of sulphurous acid gas weigh 66.89 grains, and contain 33.445 grains of sulphur; 50 cubic inches of oxygen combine with 100 cubic inches of hydrogen, or with 2.117 grains. So that, according to this analysis, sulphureted hydrogen is composed of

| Hydrogen | 2.117 or 1.000 | |----------|----------------| | Sulphur | 33.445 |

So that, according to this determination, the weight of an atom of sulphur is 15.8. This differs only .2 from the preceding, and is also within the limits of the unavoidable errors of chemical experiment. We may, therefore, consider these two methods of determining the weight of an atom of sulphur as all coinciding, and giving absolutely the same result.

By a similar mode of reasoning, we may determine, with considerable accuracy, the weight of an atom of azote, phosphorus, carbon, the metals, and almost all those bodies which at present are considered as simple. It would be tedious to give any more illustrations here. Enough, we conceive, has been said to explain the nature of the theory, and to render the manner of proceeding in determining the weights of the atoms of bodies intelligible to the reader. In a future part of this article, we shall give a table of the weights of the atoms of bodies, and state, at the same time, the documents on which these weights are founded.

It is hardly necessary to observe, how very powerfully a particular conclusion is confirmed, when we arrive at it by different processes. This advantage we have in full perfection, when we set about determining the weight of the atoms of the simple substances. In most cases, we come to the same conclusions by two, three, or four different methods. These coincidences could not exist, unless the conclusions were well founded.

With respect to the number of atoms capable of uniting together, no general rule can be given. In organic bodies, this number is often very great. Hence, probably, the reason why we cannot succeed in our attempts to form animal and vegetable bodies. In unorganic bodies, the number of atoms united is much smaller. In primary compounds, we do not recollect an example of any substance containing more than 7 atoms. These atoms are always confined to two different kinds of matter, as hydrogen and oxygen, carbon and oxygen, sulphur and oxygen, carbon and hydrogen, sulphur and phosphorus, &c. We never find three different kinds of matter united together in any primary compound, provided it does not belong to the vegetable or animal kingdom. For primary compounds of organized matter contain usually three, and sometimes four, or even five or six, different kinds of substances; as hydrogen, carbon, and oxygen; hydrogen, carbon, azote, oxygen; hydrogen, carbon, azote, iron, oxygen; hydrogen, car-

bon, azote, phosphorus, sulphur, oxygen. According to Berzelius, one of the constituents of unorganized bodies always amounts only to 1 atom. This law certainly holds in the primary compounds. Whether it holds in those compounds called salts, is not so clear. We shall be better able to judge, when we have examined the composition of the salts in a subsequent part of this article.

In organic bodies, on the contrary, none of the constituents exists of necessity to the amount of 1 atom; though this may sometimes happen from accident. Hence the reasons of the very complex nature of organic bodies, and the numerous modifications of most of them that exist.

The secondary compounds, or those bodies formed by the union of primary compounds with each other, are of a much more complex nature. They frequently contain three or four different ingredients united together; as, for example, alum, tartar emetic, Rochelle salt. Now, as each of these constituents is composed of several atoms, it is obvious, that the composition of the secondary compounds is much more complicated than that of the primary compounds. It will be convenient, on that account, to consider each separately.

II. Weight of the Atoms of the Simple Substances.

Before we can draw up a table of the relative weights of the simple substances, we must fix upon some one whose atom shall be represented by unity. Mr Dalton has made choice of hydrogen for this purpose, because it is the lightest of all known bodies. Sir Humphry Davy has followed his example; but he has doubled the weight of an atom of oxygen, and of most other bodies, by the arbitrary supposition, that water is composed of 2 atoms of hydrogen, and 1 atom of oxygen. Dr Wollaston, Professor Berzelius, and Dr Thomson, have adopted oxygen as the most convenient unit; nor can there be any hesitation in embracing their plans. Oxygen is, in fact, the substance by means of which the weights of the atoms of almost all other bodies are determined. It enters into a much greater number of combinations than any other known body. Hence, if we denote the weight of its atom by 1, a considerable convenience will result to the practical chemist. We shall exhibit the weights of the atoms of the simple bodies, under the form of tables, and we shall state at the bottom of each, the documents by means of which these weights were obtained.

1. Simple Supporters.

| Substances | Weight of Atom | |------------|---------------| | Oxygen | 1.000 | | Chlorine | 4.500 | | Iodine | 15.621 |

Oxygen having been chosen for the unit, we have already stated our reasons for the weight assigned. The weight is arbitrary, and adopted merely for the sake of convenience.

Chlorous oxide (the euchlorine of Davy) is composed of 2 volumes of chlorine gas, and 1 volume of The specific gravity of chlorine gas is 2.483; that of oxygen gas 1.104. If we suppose chlorous acid to be composed of 1 atom chlorine, and 1 atom oxygen, a supposition which gives the simplest view of the composition of the chlorides, we have for the weight of an atom of chlorine this proportion, \( \frac{1.104}{2.483} = \frac{1}{2} \) or 0.498 = the weight of an atom of chlorine.

The weight of an atom of iodine is the result of the experiments of Gay-Lussac, and may be considered as nearly correct. But we could not explain, in this place, the manner of conducting the experiments by which it was ascertained, without entering into details respecting the properties of this newly discovered substance that would be quite foreign to the present article.

### 2. Simple Combustibles

| Substances | Weight of an Atom | |------------|------------------| | Hydrogen | 0.125 | | Boron | 0.733 | | Carbon | 0.750 | | Phosphorus | 1.625 | | Azote | 1.750 | | Sulphur | 2.000 | | Silicon | 2.000 |

The method of determining the weight of an atom of hydrogen has been explained in a preceding part of this article. It was there shown from the composition of water, that the weight of an atom of hydrogen is to that of an atom of oxygen as 1 to 7.56. Now, \( \frac{7.56}{1} : 1 : 0.125 = \text{weight of an atom of hydrogen} \). We have given 0.125 in the table, as the weight of an atom of hydrogen, because we consider it as established by a paper in the *Annals of Philosophy*, Vol. VI. p. 322, that the specific gravity of oxygen is 16 times that of hydrogen.

The experiments of Davy on the combustion of boron in oxygen gas, and those of Berzelius on the borates (*Annals of Philosophy*, Vol. III. p. 66), have shown that boracic acid is composed of 100 oxygen + 36.649 boron. The resemblance between boron and carbon is so great, that it is very probable they will be analogous to each other in the proportion of oxygen with which they combine. Now, we shall see afterwards that carbonic acid is a compound of 1 atom carbon, and 2 atoms oxygen. Supposing boracic acid to be a compound of 1 atom boron, and 2 atoms oxygen, to determine the weight of an atom of boron, we have this proportion \( \frac{100}{2} : 36.649 : : 1 : 0.733 = \text{weight of an atom of boron} \).

When charcoal is burnt in oxygen gas, the bulk of the gas is not altered; it is merely converted into carbonic acid gas. Therefore, if we subtract the specific gravity of oxygen gas from that of carbonic acid gas, the remainder denotes the quantity of carbon in carbonic united with the weight of oxygen indicated by the specific gravity. Now,

\[ \begin{align*} \text{Grains.} \\ 100 \text{ cubic inches of carbonic acid weigh } & 46.313 \\ 100 \text{ oxygen gas } & 33.672 \\ \text{Remainder, } & 12.641 \end{align*} \]

Therefore carbonic acid is composed of

\[ \begin{align*} \text{Carbon} & - 12.641 \text{ or } 27.29 \\ \text{Oxygen} & - 33.672 \text{ or } 72.71 \\ & 100. \end{align*} \]

We shall see afterwards that carbonic acid is composed of 1 atom carbon + 2 atoms oxygen. Therefore, to find the weight of an atom of carbon, we have this proportion, \( \frac{72.71}{2} : 27.29 : : 1 : 0.751 = \text{weight of an atom of carbon} \).

The writer of this article has ascertained, by experiments which it would be too tedious to adduce here, that phosphoric acid is composed of 100 phosphorus + 123 oxygen, and that it consists of 1 atom phosphorus + 2 atoms oxygen. Hence the weight of an atom of phosphorus is 1.625.

The determination of the weight of an atom of azote is attended with considerable difficulty. We have no doubt, that it is in reality a compound. But whether it consists of hydrogen united to oxygen, as is the opinion of Mr Miers, or of nitricum and oxygen, as Berzelius believes, has not yet been decided by satisfactory experiments. We must, therefore, in the present state of our knowledge, allow it a place among the undecomposed bodies. It scarcely belongs to the combustibles, as it does not in any case exhibit the phenomena of combustions. But we give it a place among them for want of any better situation under which it could be arranged. Azote combines with four proportions of oxygen, and constitutes four compounds, all of which have been analyzed with care. If we denominate the quantity of oxygen that unites with 100 parts of azote by \( x \), the following table will exhibit the constituents of these compounds.

\[ \begin{align*} \text{Nitrous oxide} & . 100 \text{ azote } + 1 x \\ \text{Nitrous gas} & . 100 \text{ azote } + 2 x \\ \text{Nitrous acid} & . 100 \text{ azote } + 3 x \\ \text{Nitric acid} & . 100 \text{ azote } + 5 x \end{align*} \]

From this table there can be no doubt, that \( x \) represents an atom of oxygen. Hence we have only to ascertain the value of \( x \); to be able to determine the weight of an atom of azote. Now nitrous oxide, nitrous gas, and nitric acid, have been analyzed with care, and the number in the table is the mean resulting from these analyses.

It has been ascertained by many and careful experiments, that sulphurous and sulphuric acids are composed as follows: Hence we see, that the first contains two atoms of oxygen, and the second three atoms of oxygen, combined each with one atom of sulphur. Therefore for the weight of an atom of sulphur, we have this proportion \( \frac{100}{2} : 100 : : 1 : 2 = \text{weight of an atom of sulphur.} \)

The atom of silicon, which is a combustible substance analogous to charcoal, the base of the earth called silica, is derived from the experiments of Berzelius, for which we refer to his mineralogical essay, lately translated into English, as it would not be possible to make these experiments intelligible here, without entering into details, quite inconsistent with the nature of this article. These details will probably find a place under the article Chemistry, to which therefore we refer the reader.

### 3. Metals.

| Weight of an Atom | |-------------------| | 11 Aluminum | 1.125 | | 12 Ammonium | 1.125 | | 13 Magnesium | 1.500 | | 14 Calcium | 2.625 | | 15 Tellurium | 4.027 | | 16 Zinc | 4.095 | | 17 Zirconium | 4.625 | | 18 Chromium | 4.750 | | 19 Potassium | 5.000 | | 20 Arsenic | 6.000 | | 21 Sodium | 5.875 | | 22 Strontium | 5.500 | | 23 Molybdenum | 6.013 | | 24 Glucinum | 6.833 | | 25 Manganese | 7.125 | | 26 Iron | 7.143 | | 27 Nickel | 7.305 | | 28 Cobalt | 7.326 | | 29 Yttrium | 7.375 | | 30 Copper | 8.000 | | 31 Barytium | 8.750 | | 32 Bismuth | 9.000 | | 33 Antimony | 11.250 | | 34 Cerium | 11.500 | | 35 Uranium | 12.000 | | 36 Tungsten | 12.125 | | 37 Platinum | 12.125 | | 38 Silver | 13.750 | | 39 Palladium | 14.075 | | 40 Tin | 14.750 | | 41 Rhodium | 15.000 | | 42 Titanium | 18.000 | | 43 Gold | 24.875 | | 44 Mercury | 25.000 | | 45 Lead | 26.000 |

In the preceding table we have been obliged to omit the weight of an atom of three metals, namely, osmium, iridium, and columbium, because we are at present unacquainted with the analyses of their oxides, and of all the compounds which they form. The weights of the atoms of the metals inserted, have been taken from the best documents at present in possession of chemists. But we are far from supposing that they are all correct. Some have little better than analogical reasoning in their favour, while the weight of others is founded upon a law discovered by Berzelius relative to the salts; namely, that the oxygen in the acid of a neutral salt, is always a multiple of the oxygen in the base, by 2, 3, 4, 5, &c. and this number is always constant for every particular acid.

The weight of an atom of aluminum is calculated from the composition of alum. This salt is a combination of sulphate of potash and sulphate of alumina. Sulphuric acid is saturated by a quantity of base containing one third of the oxygen in the acid. Knowing the analysis of the salt, we obtain from this law the quantity of oxygen in a given weight of alumina. If we suppose alumina a compound of 1 atom aluminum, and 1 atom oxygen, the weight of an atom of aluminum will be as in the table. This hypothesis, however, is quite arbitrary. But it seems better to be satisfied with it at present, because we are not in possession of any data to enable us to reason on the subject. The weight of the atoms of the bases of the alkaline earths, and earths proper, were ascertained by a similar process of reasoning. They are smaller than the weights of those atoms assigned by Berzelius, because he conceives them to be united with more than one atom of oxygen. We have given, however, in the table, the weight of an atom of glucinium, as determined by Berzelius in a set of new experiments, which he has published as an appendix to his treatise on mineralogy, above referred to. The weights of the atom of potassium and sodium, are obtained from the experiments of Davy, and Gay-Lussac, and Thenard. It appears that potash is a compound of 100 potassium, and 20 oxygen, and peroxide of potassium of 100 potassium, and 60 oxygen. Hence the weight of an atom of potassium is obvious. Soda is composed of 100 sodium and 34.1 oxygen; peroxide of sodium of 100 sodium, and 51.1. Now 34.1 is to 51.1 as 2:3. Hence soda must contain 2 atoms of oxygen. For its weight therefore we have this proportion \( \frac{34.1}{2} : 100 : : 1 : 5.882 = \text{weight of an atom of sodium.} \)

Ammonium is a problematic substance which never yet has been obtained in a separate state. But when a globule of mercury is moistened with ammonia, and acted on by a galvanic battery, the mercury is converted into an amalgam of the consistence of butter, and not the fourth part of its original specific gravity. This fact cannot be explained any other way than by supposing ammonia to be a compound of oxygen, and an unknown metallic base which amalgamates with the mercury in the galvanic experiment. Analogy comes strongly in favour of this opinion. We are acquainted with nearly fifty bases which have the property of neutralizing acids, of which ammonia is one. Now it is known, that all the rest contain oxygen as a constituent. Hence, it would be singular if ammonia alone, of all the bases, should constitute an exception to a general rule. There seems to be little doubt, that sal-ammoniac is a compound of chlorine and ammonium. This salt is formed by the union of equal volumes of muriatic acid gas, and ammoniacal gas. Hence, it consists of

Muriatic acid - - 100 Ammonia - - 46.178

By weight - - 146.178

Now, muriatic acid is a compound of 75.731 chlorine, and 2.23 hydrogen. Therefore, 100 muriatic acid contain 2.86 hydrogen. We must suppose this hydrogen to find in the ammonia a quantity of oxygen capable of converting it into water. Now, 2.86 hydrogen require 21% oxygen to convert them into water. Therefore ammonia must be a compound of

Ammonium 24.511 or 100. Oxygen . . 21.666 88.39.

Now, if ammonia be composed of 1 atom oxygen, and 1 atom ammonium, we have, for finding the weight of an atom of ammonium, this proportion:

\[ \frac{21.666}{24.511} : 1 : 1.149 = \text{weight of an atom of ammonium.} \]

The weight of an atom of zinc is founded on Dr Thomson's analysis of Blende (Annals of Philosophy, IV. 89). It agrees almost exactly with the analysis of oxide of zinc by Berzelius. But Berzelius doubles the weight of an atom of zinc, by supposing the existence of another oxide, containing less oxygen than the white oxide. But as this oxide has never been obtained nor examined by any one, we have no right to suppose its existence. The weight of an atom of tellurium is derived from the experiments of Berzelius. Its oxide, according to him, is composed of 100 tellurium + 24.83 oxygen.

Berzelius (Annals of Philosophy, III. 101.) has shown that there are three oxides of chromium, the green, the brown, and chromic acid. The first contains 100 metal + 42.37 oxygen, while the acid is composed of 100 metal + 84.74 oxygen. These numbers are to each other as 1 to 2. As the brown oxide is intermediate, we must suppose the green oxide a compound of 1 atom metal + 2 atoms oxygen; the brown of 1 atom metal + 3 oxygen; and the acid of 1 atom metal + 4 atoms oxygen. This gives us the weight of an atom of chromium, stated in the table. But it would not be surprising if the brown oxide were merely a mixture or combination of the green oxide and chromic acid. On that supposition the weight of an atom of chromium would be only 2.375.

Berzelius has published an elaborate set of experiments on arsenic (Annals of Philosophy, III. 93.), in which he endeavours to prove that there are four oxides of arsenic, namely, 1. The black oxide, composed of 1 atom metal + 1 atom oxygen. 2. Salpable oxide of arsenic, composed of 1 atom metal + 3 atoms oxygen. 3. Arsenious acid, composed of 1 atom metal + 4 atoms oxygen; and, 4. Arsenic acid, composed of 1 atom metal + 6 atoms oxygen. But his experiments are too complicated, and his reasoning too hypothetical, to produce conviction; especially as they appear inconsistent with the much simpler experiments of Proust, Thenard, Rose, Bucholz, and Thomson, from which it appears that 100 arsenic, when converted into arsenic acid, becomes 152.4. It was from this experiment of Dr Thomson that the weight of an atom of arsenic in the table was derived.

The weight of an atom of molybdenum is deduced from the experiments of Bucholz. He found two oxides of molybdenum, the blue and the white, composed of 100 metal united with 34 and 50 oxygen. Now 34 is to 50 as 2 to 3. Hence the first is a deuterioxide, and we have for the weight of molybdenum this proportion \( \frac{34}{2} : 100 :: 1 : 6.013 = \text{weight of an atom of molybdenum.} \)

There are four oxides of manganese, the green, the olive, the brown, and the black. The composition of these oxides is as follows:

| Oxide | Metal | Oxygen | |-------|-------|--------| | 1st | 100 | 14.0533| | 2d | 100 | 28.107 | | 3d | 100 | 42.16 | | 4th | 100 | 56.213 |

Now, the quantities of oxygen in these oxides are to each other as the numbers 1, 2, 3, 4. Hence the first is a protoxide; and for the weight of an atom of manganese we have this proportion \( 14.0533 : 100 :: 1 : 6.833 = \text{weight of an atom of manganese.} \)

Iron forms two oxides, the black and the red. The first of these, according to the experiments of Dr Thomson, is composed of 100 metal + 28 oxygen, the second of 100 metal + 42 oxygen. Now 28 is to 42 as 2 to 3. Hence \( \frac{28}{2} : 100 :: 1 : 7.143 = \text{weight of an atom of iron.} \)

Nickel and cobalt agree with iron in being magnetic. They agree, likewise, in the number of oxides which they form, and almost in the proportion of oxygen with which they unite. Probably future experiments will show these proportions to be absolutely identical. At present it is believed, that the two oxides of nickel are composed as follows:

| Oxide | Nickel | Oxygen | |-------|--------|--------| | 1st | 100 | 27.6 | | 2d | 100 | 41 |

while those of cobalt consist of

| Oxide | Cobalt | Oxygen | |-------|--------|--------| | 1st | 100 | 27.3 | | 2d | 100 | 40.95 |

These slight differences in the proportion of oxygen occasion the difference between the weights of the atoms of these metals and that of iron.

There are two oxides of copper, the red and the black, composed of Hence \(12.5 : 100 :: 1 : 8\) = weight of an atom of copper.

Bismuth forms only one oxide. A mean of the experiments of Thomson, Lagerhjelm, and John Davy, gives its composition 100 metal + 11.229. Hence the number in the table.

There is no metal so difficult to experiment on as antimony. Hence, notwithstanding the laborious efforts of many distinguished chemists, considerable uncertainty still exists, both respecting the number and the composition of its various oxides. Proust makes only two oxides of antimony: Thenard makes six, and Berzelius makes four. We think it safer, in the present state of our knowledge, to deduce the weight of an atom of antimony from the sulphuret, which can be analyzed with accuracy, than from the oxides, respecting the constitution of which considerable doubts still remain. From the experiments of Dr Thomson (Annals of Philosophy, IV. 99.), it appears that sulphuret of antimony is composed of 100 metal + 35.556 sulphur. Now, supposing it a compound of 1 atom metal + 2 atoms sulphur, an atom of antimony will weigh 11.249.

The weight of an atom of cerium is founded on the experiments of Hisinger, which may be seen in the Annals of Philosophy, IV. 355.

Bueholz has shown that there are two oxides of uranium, the black and the yellow; the first composed of 100 metal + 8.333 oxygen, the second of 100 metal + 25 oxygen. These numbers are to each other as 1 to 3. Hence 8.333 : 100 :: 1 : 12 = weight of an atom of uranium.

Berzelius has shown (Annals of Philosophy, III. 244), that there are two oxides of tungsten, brown oxide, and tungstic acid. The first is composed of 100 metal + 16.5 oxygen, the second of 100 metal + 24.75 oxygen. Now 16.5 is to 24.75 as 2 to 3.

Therefore \(\frac{16.5}{2} : 100 :: 1 : 12 : 121 =\) weight of an atom of tungsten.

Berzelius obtained two oxides of platinum. The first composed of 100 metal + 8.287 oxygen, the second of 100 metal + 16.36. Now 8.287 is to 16.36 as 1 to 2. Therefore 8.287 : 100 :: 1 : 12.161 = weight of an atom of platinum.

Silver forms only one oxide, and it parts with its oxygen so easily, that it is very difficult to determine the proportion of oxygen with which it unites. The number in the table is derived from horn-silver, which is known to be a compound of 100 chlorine and 304.89 silver. This gives us the weight of an atom of silver 13.714, and the oxide of silver, a compound of 100 silver + 7.291 oxygen.

Berzelius could form only one oxide of palladium. It was brown, and composed of 100 metal + 14.209 oxygen. If we suppose this a deutoxide, which is probable, then we have \(\frac{14.209}{2} : 100 :: 1 : 14.075 =\) weight of an atom of palladium.

Proust first proved that tin forms three oxides. According to Berzelius, the first oxide is composed of 100 metal + 13.6 oxygen, and the peroxide of 100 metal + 27.2. As there is an intermediate oxide, we cannot avoid concluding that the first oxide is a compound of 1 atom metal + 2 atoms oxygen. For the oxygen in the peroxide is just double that in the first oxide. Therefore for an atom of tin we have this proportion, \(\frac{13.6}{2} : 100 :: 1 : 14.705 =\) weight of an atom of tin.

Berzelius has shown that there are three oxides of rhodium. The first, constituting the base of the muriate, is composed of 100 metal + 6.71 oxygen. The second of a fleck brown colour, obtained by heat, is composed of 100 metal + 13.42 oxygen. The third is red, and is obtained by precipitation from the soda-muriate. It contains more oxygen than either of the other two (Annals of Philosophy, III. 252.). We have therefore 6.71 : 100 :: 1 : 14.903 = weight of an atom of rhodium.

The weight of an atom of titanium, given in the table, is not much to be depended on. It is the number given by Berzelius from an experiment of Richter (Annals of Philosophy, III. 251.), and is merely inserted for want of better data.

To the indefatigable industry and address of Berzelius, we are indebted for our knowledge of the composition of the oxides of gold. His mode of proceeding was not quite beyond the reach of objection, though perhaps, upon the whole, the best that he could have had recourse to, with the prospect of an answer to the problem which he was investigating. A determinate quantity of gold was dissolved in nitromuriatic acid, and thus converted into yellow oxide or peroxide of gold. He ascertained how much mercury was necessary to precipitate this gold in the metallic state. This portion of mercury united with all the oxygen of the gold. Now, as the composition of oxide of mercury is known, it was easy to determine how much oxygen had been united with the gold. By exposing the muriate of gold to heat, its nature was altered, and another oxide of gold formed, containing less oxygen. When water is poured upon this substance, two-thirds of the gold are reduced, and the other third converted into yellow oxide. Hence this new oxide contained only the third part of the oxygen in the yellow oxide. The yellow oxide was composed of 100 gold + 12.077 oxygen. Hence, the protioxide consisted of 100 gold + 4.026 oxygen. Now 4.026 : 100 :: 1 : 24.338 = weight of an atom of gold.

The weight of an atom of mercury, is taken from the experiments of Thenard, and Fourcroy, and Sefstrom. The mean of these give us black oxide of mercury, composed of 100 mercury + 4 oxygen, and red oxide of 100 mercury + 8 oxygen.

There are three oxides of lead, the yellow, the red, and the brown; the first composed of 100 metal + 7.7 oxygen, the second of 100 metal + 11.55 oxygen, and the third of 100 metal + 15.4 oxygen. Now 7.7, 11.55, and 15.4, are to each other as the numbers 2, 3, 4. Therefore, the yellow oxide con- tains two atoms of oxygen, and we have \( \frac{7.7}{2} : 100 :: 1 : 1 \).

\( 25.974 = \) weight of an atom of lead.

Such are the weights of the atoms of the simple substances, and such the documents upon which these weights are founded. The importance of these weights will best appear by giving a view of the compounds which these bodies form with each other.

### III. Weight of the Integrant Particles of Primary Compounds.

The primary compounds being very numerous, we shall, for the sake of distinctness, subdivide them under different heads.

#### 1. Compounds of the Simple Combustibles with Oxygen.

| Substances | Number of Atoms | Weight of an Integrant Particle | |---------------------|-----------------|--------------------------------| | Water | 1a + 1o | 1'125 | | Boracic acid | 1b + 2a | 2'733 | | Carbonic oxide | 1c + 1o | 1'750 | | Carbonic acid | 1c + 2o | 2'750 | | Phosphorous acid | 1p + 1o | 2'625 | | Phosphoric acid | 1p + 2o | 3'625 | | Nitrous oxide | 1a + 1o | 2'750 | | Nitrous gas | 1a + 2o | 3'750 | | Nitrous acid | 1a + 3o | 4'750 | | Nitric acid | 1a + 3o | 6'750 | | Sulphurous acid | 1s + 2o | 4'000 | | Sulphuric acid | 1s + 3o | 5'000 | | Silica | 1s + 2o | 6'000 |

Water is composed of two measures of hydrogen gas, united to one measure of oxygen. The reasons for considering it as a compound of 1 atom hydrogen, and 1 atom oxygen, have been already stated.

Boracic acid is conceived to contain two atoms of oxygen, because this is the case with carbonic acid, and the analogy between carbon and boron is very striking.

If 100 measures of carbonic oxide be mixed with 50 measures of oxygen gas, and the mixture be fired by electricity, the result will be 100 measures of carbonic acid. If charcoal be burnt on 100 measures of oxygen gas, the bulk is not altered, but the oxygen is changed into carbonic acid. From these facts, it is obvious that the oxygen in carbonic oxide, combined with a given quantity of carbon, is just half of that in carbonic acid combined with the same quantity of carbon. Hence the first must be a compound of 1 atom carbon + 1 atom oxygen; the second of 1 atom carbon + 2 atoms oxygen. The weights in the table correspond exactly with the analysis of these two substances.

Several of the compounds of azote and oxygen have been analyzed with great care; namely, nitrous oxide, nitrous gas, and nitric acid; and the results coincide with the numbers given in the table. Nitrous acid is merely given from theory. It has not hitherto been possible to analyze it.

Sulphurous acid was first analyzed by Dr Thom-

son, who showed that it contained two-thirds of the oxygen in sulphuric acid. It seems demonstrated by the experiments of Klaproth, Berzelius, &c. that sulphuric acid is a compound of 100 sulphur + 150 oxygen. Therefore sulphurous acid must be composed of 100 sulphur + 100 oxygen, and the composition of both must be as in the table.

The composition of silica is taken from the experiments and calculations of Berzelius, and depends chiefly upon analogical reasoning.

#### 2. Compounds of the Simple Combustibles with each other.

| Number of Atoms | Weight of an Integrant Particle | |-----------------|--------------------------------| | Olefiant gas | 1c + 1h | 0.875 | | Carbureted hydrogen | 1c + 2h | 1.000 | | Hydrophosphoric gas | 1p + 2h | 1.875 | | Phosphureted hydrogen | 1p + 4h | 2.125 | | Ammonia | 1a + 3h | 2.125 | | Sulphureted hydrogen | 1s + 1h | 2.125 | | Sulphuret of carbon | 1c + 2s | 4.750 |

Olefiant gas requires for complete combustion three times its bulk of oxygen gas, and forms twice its bulk of carbonic oxide; carbureted hydrogen requires twice its bulk of oxygen, and forms its own bulk of carbonic acid. Hence, it is obvious, that supposing the quantity of hydrogen the same in both, the carbon in olefiant gas, is just double that in carbureted hydrogen. We might have conceived carbureted hydrogen to be a compound of 1 atom hydrogen, and 1 atom carbon, and olefiant gas of 1 atom hydrogen, and 2 atoms carbon. But the reasons assigned by Mr Dalton, have induced us to adopt the composition in the table, which comes in fact to the very same thing.

The two gases which are composed of phosphorus and hydrogen, have not been hitherto analyzed. But it appears from Davy that the specific gravity of the first is double that of the second. Davy states his hydrophosphoric gas to be composed of 100 hydrogen by weight, and 489.56 phosphorus. The numbers in the table are founded on these data; though they do not quite correspond.

Ammonia, by electricity, may be resolved into 3 measures of hydrogen, and 1 measure of azote. 300 cubic inches of hydrogen gas weigh 6.69 grains, and 100 cubic inches of azotic gas weigh 29.56 grams. Now \( \frac{6.69}{29.56} : 0.132 \times 3 : 1.704 = \) weight of an atom of azote very nearly.

We have given the data upon which the composition of sulphureted hydrogen is founded in a preceding part of this article. It is therefore unnecessary to repeat them here.

Sulphuret of carbon was analyzed by Berzelius and Marcat, who found it a compound of 84.85 sulphur, and 15.17 carbon. The composition of this substance, as given in the table, supposes it a compound of 84.2 sulphur + 15.8 carbon; numbers which may be considered as identical with the preceding, since the difference is within the limits of the unavoidable errors of experiment. This table contains all the metallic oxides at present known, which are 62 in number. We have omitted a few which Thenard has admitted into his *System of Chemistry*, just published; but which do not seem to be supported by sufficiently strong evidence, to induce us to admit their existence. The same remark applies to some of the oxides of Gay-Lussac and Thenard; and, perhaps, also to some of the oxides of Berzelius. In short, we have admitted no primary compound into our tables, the existence of which is not demonstrated by the clearest evidence. For the loose admission of hypothetical bodies seems to us injurious, rather than beneficial to the progress of the science.

Alumina, Magnesia, Lime, Zirconia, Strontian, Yttria, Barytes, are supposed compounds of 1 atom base, and 1 atom oxygen; because we do not know any other compounds which these bases form with oxygen; and, therefore, have no data to reason on the subject. Their composition has been calculated from the composition of the sulphates of alumina, magnesia, lime, zirconia, strontian, yttria, barytes, the constitution of which will be shown in a subsequent part of this article. The calculation was made, on the supposition, that the quantity of each base which saturates a given weight of sulphuric acid, contains just one-third of the oxygen in the acid: a law which holds, in every case, in which we have it in our power to verify it.

Ammonia, in the present state of the sciences, is an enigmatical substance. We have given before a view of its composition, on the supposition, that it is a compound of hydrogen and azote. Here, its composition is seen on the supposition that it is a compound of ammonium and oxygen. Both analyses seem demonstrated. The seeming inconsistency will vanish, whenever it can be shown that azote is a compound of hydrogen and oxygen.

Potash, peroxide of potash, soda, peroxide of soda, are given according to the experiments of Davy, and Gay-Lussac, and Thenard. The French chemists admit an oxide of potassium and sodium, containing each less oxygen than exists in potash and soda. But the existence of these supposed oxides is not made out in a satisfactory manner. For anything that appears to the contrary, their oxides may be mixtures of potash and potassium, soda and sodium.

In stating the weights of the atoms of the metals, we have noticed the most accurate experiments hitherto made, to determine the number and composition of the different metallic oxides; and it does not seem necessary to resume the subject here. If the reader be at the trouble to make the trial, he will find, that the numbers in the table agree very correctly with the best results of the analysis of the metallic oxides hitherto obtained. Hence there can be no doubt, that at least a considerable part of the table is accurate. According to the experiments of Dr Thomson (Annals of Philosophy, IV. 94.), sulphuret of zinc is composed of 100 metal + 48.84 sulphur. Hence the numbers in the table which have been derived from that analysis.

The composition of sulphuret of tellurium is derived from an experiment of Davy. He found, that tellurium united by fusion with nearly its own weight of sulphur. Hence it ought to combine with two atoms. Analogy, however, is against this conclusion. In all other cases, when a metal forms only one oxide, its sulphuret contains only one atom of sulphur. Hence the composition of sulphuret of tellurium, as stated in the table, is very doubtful.

The composition of sulphuret of potassium is derived from the experiments of Davy. Sulphuret of sodium is stated from the following analogy, having never been analyzed. Berzelius has shown, that all the sulphurets, when treated with nitric acid, are converted into neutral sulphates, without any redundancy of acid or base. Therefore, sulphuret of sodium, by this treatment, would be converted into sulphate of soda. But sulphate of soda is composed of 2 integrant particles of sulphuric acid, and 1 integrant particle of soda. Therefore, sulphuret of sodium must be composed of 2 atoms sulphur, and 1 atom sodium.

From the experiments of Laugier it appears, that sulphuret of arsenic is composed of 42 sulphur + 58 arsenic. According to him, it is often mixed with an excess of arsenic, which occasions the apparent variability in its composition. The experiments of Proust and Thenard do not agree well with those of Laugier. Hence the subject cannot be considered as completely cleared up.

Sulphuret of manganese is derived from an experiment of Vauquelin, who found that 74.8 manganese united with 25.5 of sulphur by heat.

Bucholz found sulphuret of molybdenum composed of 60 molybdenum + 40 sulphur. Hence its constitution as stated in the table.

The two sulphurets of iron were first analyzed by Hatchett, and the table is conformable to the results of that analysis. We consider the late speculations of Stromeyer on magnetic pysites as erroneous.

Nickel and cobalt have so close an analogy to iron in their combinations, that it is reasonable to expect that the sulphurets will also correspond. Hence the reason for considering these sulphurets as compounds of one atom metal and two atoms sulphur. This statement likewise approaches nearest to the results obtained by Proust.

It has been ascertained that 100 copper combine with 25 sulphur. This coincides exactly with the statement in the table.

John Davy has shown that 67.5 bismuth unite with 15.08 sulphur. Lagerhjelm found that 100 bismuth unite with 22.52 sulphur. Both of these statements coincide sufficiently with the table.

From the analysis of sulphuret of antimony by Dr Thomson (Annals of Philosophy, IV. 99.), we learn, that it is a compound of 100 metal + 35.556 sulphur. With this analysis the statement in the table exactly coincides. Berzelius informs us, that 100 platinum combine with 32.8 sulphur. Now \( \frac{108}{32.8} : \frac{12.161}{3.098} = 2 \) atoms of sulphur very nearly.

According to Berzelius, 100 sulphur combine with 14.9 sulphur. Now \( \frac{100}{14.9} : \frac{13.714}{2.043} = 1 \) atom of sulphur very nearly.

Sulphuret of palladium has not been analyzed; but, as palladium forms only one oxide, we may, without much risk of error, consider the sulphuret as a compound of 1 atom metal and 1 atom sulphur.

From the experiments of Dr Davy it appears, that mosaic gold contains just double the quantity of sulphuret in common sulphuret of tin. According to the experiments of the same chemist, protosulphuret of tin is a compound of 55 tin + 15 sulphur. Now \( \frac{55}{15} : \frac{14.705}{4.015} = 2 \) atoms of sulphur very nearly.

Oberkampf has shown, that the sulphuret of gold is composed of 100 gold + 24.39 sulphur. Now \( \frac{100}{24.39} : \frac{24.838}{6.057} = 3 \) atoms of sulphur very nearly.

According to the experiments of Sefstrom, black sulphuret of mercury is composed of 100 metal + 8.005 sulphur; and red sulphuret of 100 metal + 16.01 sulphur. Now \( \frac{100}{8.005} : \frac{25}{2.001} \) and \( \frac{100}{16.001} : \frac{25}{4.002} \). But 2.001 and 4.002 are obviously equal respectively to 1 atom and 2 atoms of sulphur.

Galena is a compound of 100 lead + 15.42 sulphur. Now \( \frac{100}{15.42} : \frac{25.974}{4.005} = 2 \) atoms of sulphur. Sir John Sinclair found a variety of galena in Caithness, which burned with a blue flame when held to a candle. It was analyzed by Dr Thomson, and found to contain double the quantity of sulphur in common galena. Hence it was a compound of 1 atom lead and 3 atoms sulphur. This variety must be scarce, for it has never been noticed by mineralogists.

Such are the sulphurets with the composition of which we are at present acquainted. A good number are of necessity omitted, because they have never been analyzed by chemists. It would be easy to state the composition of most of these from theory. But we do not see any advantage that would result to the science from such theoretical statements, unsupported by experiment.

Neither the phosphurets nor carburets have been examined with such precision as to enable us to exhibit their composition in a table. It is probable that the phosphurets are analogous to sulphurets. But certainly no such analogy holds with the carburets and silicarrets; both of which exist, though they have been hitherto too slightly examined to give us any exact notions of their constitution. Carbon seems to combine with a great number of atoms of iron, in cast iron, and with a still greater number in steel.

5. Organic Bodies.

Hitherto the only correct analysis of organic bodies are those made by Berzelius, and published in the fifth volume of the Annals of Philosophy. As these experiments are so recent as not to be generally known, we shall state in the table the results of each analysis, and likewise the number of atoms of which the body is composed.

| Constituents | Atoms | Weight | Atomic Theory | |--------------|-------|--------|---------------| | Citric acid | oxygen | 55.096 | 2 | | | carbon | 41.270 | 2 | | | hydrogen | 3.634 | 1 | | Tartaric acid| oxygen | 60.213 | 10 | | | carbon | 35.980 | 8 | | | hydrogen | 3.807 | 5 | | Oxalic acid | oxygen | 66.534 | 12 | | | carbon | 33.292 | 8 | | | hydrogen | 0.244 | 1 | | Succinic acid| oxygen | 47.888 | 3 | | | carbon | 47.600 | 4 | | | hydrogen | 4.512 | 2 | | Acetic acid | oxygen | 46.832 | 3 | | | carbon | 46.833 | 4 | | | hydrogen | 6.35 | 3 | | Gallic acid | oxygen | 38.02 | 3 | | | carbon | 56.96 | 6 | | | hydrogen | 5.02 | 5 | | Saccharic acid| oxygen | 61.465 | 8 | | | carbon | 33.430 | 6 | | | hydrogen | 5.105 | 5 | | Benzoic acid | oxygen | 20.43 | 3 | | | carbon | 74.41 | 15 | | | hydrogen | 5.16 | 6 | | Tannin from Nutgalls | oxygen | 44.654 | 4 | | | carbon | 51.160 | 6 | | | hydrogen | 4.186 | 3 | | Common sugar | oxygen | 49.015 | 20 | | | carbon | 44.200 | 24 | | | hydrogen | 6.785 | 21 | | Sugar of milk| oxygen | 53.359 | 8 | | | carbon | 39.474 | 10 | | | hydrogen | 7.167 | 8 | | Gum Arabic | oxygen | 51.306 | 12 | | | carbon | 41.906 | 13 | | | hydrogen | 6.788 | 12 |

100,000 37 For the way in which these experiments were made, we refer to the original paper of Berzelius, quoted above. We have no doubt that the results are very near approximations to the truth, though some slight corrections may perhaps be necessary. The great number of atoms that enter into the composition of some of these bodies is very remarkable. We would find it difficult to explain the nature of such combinations, without having recourse to some hypothesis respecting the nature of elasticity. Twelve atoms of oxygen, and 13 of hydrogen, in a compound possessed of so little elasticity as potato starch, is very extraordinary. The old opinion of Dr Black and his contemporaries was, that bodies owe their elasticity to a quantity of heat combined with them. Of course, when the heat is separated, the elasticity disappears. Were we to suppose, in conformity with this hypothesis, that the oxygen and hydrogen which exists in organized bodies, are deprived of the heat requisite to give them the elastic form, there would be no difficulty in understanding how so many atoms unite together, and have no tendency to separate, and how the compound is so speedily decomposed by the application of heat, or by combustion.

Thus, we have gone over all the primary compounds which have been subjected to examination by chemists, without finding a single exception to the atomic theory. On the contrary, the great and unexpected elucidation which this theory throws upon the nature of chemical combinations, is highly pleasing. The simplicity of the proportions in which the atoms of unorganic matter unite, is equally delightful and astonishing, and throws a degree of grandeur around the science, hitherto conversant in minute and apparently unconnected combinations, fully equal to what distinguishes the sublimest branch of mechanical philosophy. The operations of nature are everywhere equally grand, and regulated by general principles equally simple and luminous, whether we examine the laws which regulate the motions of the solar system, or the minutest compound to be found in the terrestrial globe. Let us now examine the secondary compounds, that we may be able to judge how far they come under the dominion of the Atomic Theory.

IV. Composition of Secondary Compounds.

Here, our task is much more difficult, and the field in a great measure unexplored, or only partially so. The secondary compounds are very numerous, amounting to several thousands. Accurate chemical analysis cannot be dated farther back than a few years. A correct knowledge of the whole of these compounds, therefore, is not to be expected. But, from the indefatigable zeal, uncommon accuracy, and gigantic industry of Berzelius, we have it in our power to examine the composition of many genera of salts with considerable precision; supposing always that the laws which he has pointed out respecting the combination of acids and bases be precise. We shall continue the method which we adopted when treating of the primary compounds; namely, exhibit tables of the composition of the different genera of secondary compounds, and then give the documents on which these tables were founded.

1. Hydrates.

| Number of Atoms | Weight of an Integrant Particle | |-----------------|--------------------------------| | 169 Hydrate of potash | 1 p + 1 w = 7.125 | | 170 Hydrate of soda | 1 s + 2 w = 10.125 | | 171 Hydrate of lime | 1 l + 1 w = 4.750 | | 172 Hydrate of barytes | 1 b + 1 w = 10.875 | | 173 Hydrate of strontian | 1 st + 1 w = 7.725 | | 174 Hydrate of magnesia | 2 m + 1 w = 6.125 | | 175 Hydrate of alumina | 1 a + 1 w = 3.250 | | 176 Hydrate of glucina | 1 g + 1 w = 10.965 | | 177 Hydrate of yttria | 1 y + 3 w = 11.750 | | 178 Hydrate of zirconia | 1 z + 1 w = 6.750 | | 179 Hydrate of silica | 1 si + 1 w = 3.125 | | 180 Hydro-sulphuric acid, or acid of 1.85 | 1 s + 1 w = 6.125 | | 181 2d hydrate of sulphuric acid, or acid of 1.780 | 1 s + 2 w = 7.250 | | 182 3d hydrate of sulphuric acid, or acid of 1.65 | 1 s + 3 w = 8.375 | | 183 Hydronitric acid, or acid of 1.62 | 2 n + 1 w = 14.625 | | 184 2d hydrate of nitric acid, or acid of 1.54 | 1 n + 1 w = 7.875 | | 185 3d hydrate of nitric acid, or acid of 1.42 | 1 n + 2 w = 9.000 | | 186 4th hydrate of nitric acid, or acid of 1.350 | 1 n + 3 w = 10.125 | | 187 Hydrophosphorous acid | 2 p + 1 w = 6.375 | | 188 Hydrate of boracic acid | 1 b + 2 w = 4.997 | | 189 Hydrate of peroxide of copper | 1 c + 1 w = 11.125 | | 190 Hydrate of black oxide of iron | 1 i + 1 w = 10.240 | | 191 Hydrate of red oxide of iron | 1 i + 1 w = 11.240 | | 192 Hydrate of deutoxide of tin | 1 t + 1 w = 17.875 | | 193 Hydrate of peroxide of tin | 1 t + 1 w = 19.875 | | 194 Hydrate of deutoxide of nickel | 1 n + 3 w = 12.701 | | 195 Hydrate of deutoxide of cobalt | 1 c + 1 w = 11.458 | | 196 Hydrate of protoxide of manganese | 1 m + 1 w = 9.250 | | 197 Hydrate of oxide of arsenic | 1 o + 1 w = 7.125 | | 198 Hydrate of antimony | 2 a + 1 w = 31.625 | | 199 Hydrate of oxalic acid | 1 o + 12 w = 31.724 | | 200 Hydrate of tartaric acid | 1 t + 2 w = 18.932 | | 201 Hydrate of citric acid | 2 c + 3 w = 10.664 |

VOL. I. PART II. By hydrate of potash is understood caustic potash, that has been exposed to a red heat. For it is well known, that the whole water cannot be expelled from potash by heat. The statement in the table supposes this hydrate to be composed of 100 potash + 18.867 water. Now Davy, by heating potash and boracic acid together, actually separated between 17 and 18 water. Berzelius obtained 16 per cent., which is very near the exact quantity. For if our determination be correct, 100 potash ought to contain 15.872 water.

Hydrate of soda has never been analyzed. The number in the table is given from this analogy. Berzelius has shown, that the hydrates of the salifiable bases, are in fact salts, the water acting the part of an acid, and saturating the base. Now, in all the salts of soda hitherto examined, 1 integrant particle of the soda is always combined with 2 integrant particles of acid. We suppose the same thing to hold with water.

Slacked lime, which is the hydrate, according to the experiments of Lavoisier, is composed of 100 lime + 28.7 water; according to Dalton, of 100 lime + 33.333 water. The mean of these two results gives the statement in the table almost precisely.

The experiments of Berthollet and Berzelius prove, that hydrate of barytes is a compound of 100 barytes + 12.121 water. The statement in the table supposes it a compound of 100 barytes + 11.632, which coincides very nearly.

Hydrate of strontian is given merely from analogy. Crystallized strontian is a compound of 1 particle strontian + 13 particles of water.

According to Davy, hydrate of magnesia contains one-fifth of its weight of water. Hence the statement in the table. Probably there may be another hydrate composed of 1 integrant particle magnesia, and 1 integrant particle water, but which cannot be obtained in a separate state.

Wavelite, which is a native hydrate of alumina, is composed of 74 alumina + 26 water. Hence the composition of the hydrate in the table.

Hydrate of glucina has never been analyzed. According to Klaproth, hydrate of yttria is composed of 69 yttria + 31 water. According to Davy, zirconia, when in the state of a hydrate, contains more than one-fifth of its weight of water.

Hydrate of silica has never been analyzed. But silica absorbs about one-fourth of its weight of water.

The hydrate of sulphuric and nitric acids were determined by Mr Dalton. Sulphuric acid of 1.85 is the strongest that can be made. It is composed of 100 real acid + 22.64 water. The second hydrate is composed of 100 acid + 45.28 water. It freezes at the temperature of 42°. It was upon it that Mr Keir made his experiments.

The hydrate of phosphorous acid was determined by Sir Humphry Davy. See his Elements of Chemical Philosophy, p. 289.

Davy found hydrate of boracic acid composed of 57 acid + 43 water. Now, this agrees very nearly with the statement in the table.

Hydrate of peroxide of copper is obtained by precipitating the sulphate or nitrate of copper, by means of an alkali. It is a blue powder composed, according to Davy, of 9 peroxide of copper + 1 water. This agrees nearly with the statement in the table.

According to Berzelius, when iron is kept in water, a hydrate is formed, composed of 128 parts of black oxide of iron + 15.866 water. Now, this coincides almost exactly with 1 integrant particle of black oxide, 1 integrant particle of water.

Hydrate of the red oxide of iron is the orange powder obtained from persulphate of iron by an alkali. It has never been analyzed.

The hydrate of deutoxide of tin, is the white powder obtained by precipitating the recent solution of tin in muriatic acid. According to Proust, it is composed of 95 oxide + 5 water. This approaches most nearly to the statement in the table. The other hydrate of tin has never been analyzed.

The hydrate of nickel is a green powder obtained by precipitating sulphate of nickel by an alkali. According to Davy, more than one-fourth of its weight is water. This statement approaches most nearly to the supposition, that it contains 3 particles of water + 1 oxide of nickel.

The hydrates of cobalt, manganese, and arsenic, have never been analyzed. They are stated merely from analogy.

According to Berzelius, the tritoxide of antimony forms a hydrate composed of 95.22 oxide + 4.78 water. This approaches most nearly to the statement in the table; but does not agree with it.

The hydrates of oxalic acid, tartaric acid, and citric acid, were determined by Berzelius. Crystallized oxalic acid is composed of 58 acid + 42 water; crystallized tartaric acid of 58.75 acid, and + 11.25 water; and crystallized citric acid of 100 acid + 20.5 water. These proportions coincide with the statements in the table very nearly.

Thus, it appears that the composition of the hydrates, as far as they have been examined, agree perfectly with the atomic theory. We see, likewise, that the number of particles of water which enter into combination with other bodies is not confined within such narrow limits as most of the other primary compounds are. In many hydrates, only a single particle of water is found; while, in others, no fewer than 12 particles exist.

Let us now proceed to examine the salts, in order to judge how far their composition is regulated by the laws of the atomic theory.

2. Salts.

The salts are very numerous, and have been divided into as many genera as there are acids entering into their composition. Some of these genera have been carefully analyzed, while others still remain in a great measure unknown.

**Genus I.—Sulphates.**

| Number of Atoms | Weight of an Integrant Particle | |-----------------|--------------------------------| | 202 Sulphate of potash | 1 s + 1 p = 11.000 | | 203 Bisulphate of potash | 2 s + 1 p = 16.000 | Dr Thomson found sulphate of potash composed of 42.2 acid + 50.1 potash, which corresponds with the tabular statement almost exactly. Dr Wollaston has shown, that the bisulphate contains just double the acid in the sulphate.

Sulphate of soda, according to Wenzel, is composed of 100 acid + 78.32 base; according to Berzelius, of 100 acid + 79.34. The tabular statement supposes a combination of 100 acid + 78.82 soda. Now, this is the mean of the two experiments.

According to Berzelius, sulphate of ammonia is composed of 100 acid + 42.561 ammonia. Now 100:42.561::5:2.128 = weight of an integrant particle of ammonia very nearly.

According to Berzelius, sulphate of magnesia is composed of 100 acid + 50.06 magnesia. Now 100:50.06::5:2.508 = weight of an integrant particle of magnesia very nearly.

Sulphate of lime, according to the analysis of Berzelius, is composed of 100 acid + 72.41 base. Now,

\[ 100:72.41::5:3.620 = \text{weight of an integrant particle of lime almost exactly.} \]

Sulphate of barytes, according to the analysis of Berzelius, is composed of 100 acid + 194 barytes. Now 100:194::5:9.700 = weight of an atom of barytes very nearly.

The weight of an integrant particle of strontian was deduced from an accurate analysis of carbonate of strontian, which was found to consist of 29.9 carbonic acid + 70.1 strontian.

According to Berzelius's analysis, sulphate of alumina is composed of 100 acid + 42.722 alumina. It was from this analysis, that the weight of an integrant particle of alumina was determined. Of course, the statement in the table must coincide with it entirely.

The sulphates of yttria, glucina, and zirconia, have not hitherto been analyzed. Their composition is stated merely from analogy.

Alum, according to the experiments of Berzelius, is composed of

- Sulphuric acid: 34.33 - Alumina: 10.86 - Potash: 9.81 - Water: 45.00

Now, this will be found to consist of an integrant particle of sulphate of potash, combined with five integrant particles of sulphate of alumina.

Sulphate of potash and ammonia is a compound of 1 integrant particle of sulphate of potash, and 1 integrant particle of sulphate of ammonia. All the other triple salts are formed in the same way, by the combination of 1 integrant particle of 1 salt, with 1 or more integrant particles of another salt. Sulphate of potash and magnesia consists of 1 integrant particle of sulphate of potash, united with 2 integrant particles of sulphate of magnesia; sulphate of soda and ammonia is composed of 1 integrant particle of sulphate of soda, united with 6 integrant particles of sulphate of ammonia; sulphate of soda and magnesia of 1 integrant particle of sulphate of soda, and 3 integrant particles of sulphate of magnesia; and sulphate of magnesia and ammonia of 2 integrant particles of sulphate of magnesia, and 1 integrant particle of sulphate of ammonia.

As an atom of sulphuric acid weighs 5, and an atom of black oxide of copper 10, it is obvious that if sulphate of copper contains 2 particles of acid, and 1 of oxide, the weight of the two constituents must be equal. Now, according to Proust, the salt is composed of 33 acid + 32 base; according to Berzelius of 49.1 acid + 50.9 oxide. The mean of these two experiments gives us the salt composed of 100 acid + 100.317 base, which almost coincides with theory. According to Proust, the subsulphate of copper is composed of 18 acid + 68 oxide. Now 5:10 × 2 :: 18:72, this does not differ much from the analysis, if we consider the difficulty in getting the salt quite pure, and in a proper state for examination.

Sulphate of iron, according to the analysis of Ber- According to Berzelius, sulphate of silver is composed of 25.78 acid + 74.22 oxide. Now 5:14:714 :: 25.78:75.865, which corresponds with the analysis very nearly.

Lagerhjelm found sulphate of bismuth composed of 33.647 acid + 66.653 oxide. Now, this corresponds almost exactly with the statement in the table.

According to Tapputi, sulphate of nickel is composed of 100 acid + 87.266. Now, this does not differ very much from the statement in the table, which supposes the salt composed of 100 acid + 93.05 oxide.

According to the analysis of Rolhoff, sulphate of cobalt is composed of 52.11 acid + 47.89 oxide. If the statement in the table be correct, it will be composed of 51.744 acid + 48.256 oxide. Now, this may be considered as agreeing with the analysis, since the difference is within the limits of the unavoidable errors of experiment.

According to John, sulphate of manganese is composed of 100 acid + 91.326 protoxide. Dr Thomson lately analyzed this salt, and obtained for its constituents 100 acid + 71 protoxide. Now 100:71 :: 10:7.1; and 7.1 does not differ much from the weight of a particle of protoxide of manganese. The mean of the two experiments gives 8.113, which comes still nearer the true result.

Sulphate of uranium has hitherto been analyzed by Bucholz only. He makes it a compound of 22.1 acid + 77.9 peroxide. If the statement in the table be correct, as is probable, it ought to be a compound of 25 acid + 75 peroxide.

The persulphate of platinum, according to Berzelius, is composed of 41.223 acid + 58.777 peroxide. Supposing the tabular statement correct, it ought to be a compound of 41.389 acid + 58.611 peroxide. Now, this is identical with the experimental result of Berzelius.

Thus we have gone over the sulphates, the genus of salts, which has been most accurately examined, and have found the whole of them to correspond most accurately with the atomic theory. They all agree with Berzelius's law, that the oxygen in the acid is a multiple by a whole number of the oxygen in the base, except in the instance of subsulphate of iron. But perhaps his law was only meant to ap- ply to the neutral salts, and not to the subsalts or supersalts. His law, that the oxygen in the acid amounts to three times the oxygen in the base, holds in all the neutral sulphates, except the persulphate of mercury, and sulphate of uranium; and neither of these salts can be considered as analyzed with such accuracy as to lay much stress upon their composition, as stated in the table. Let us now examine the carbonates, which are likewise pretty accurately known.

**Genus II.—Carbonates.**

| Number of Atoms | Weight of an Integrant Particle | |-----------------|--------------------------------| | 237 Bicarbonate of potash | 2 c + 1 p 11.500 | | 238 Carbonate of potash | 1 c + 1 p 8.750 | | 239 Supercarbonate of soda | 3 c + 1 s 16.125 | | 240 Carbonate of soda | 2 c + 1 s 13.375 | | 241 Bicarbonate of ammonia | 2 c + 1 a 7.625 | | 242 Carbonate of ammonia | 1 c + 1 a 4.875 | | 243 Carbonate of lime | 1 c + 1 l 6.375 | | 244 Carbonate of barytes | 1 c + 1 b 12.500 | | 245 Carbonate of strontian | 1 c + 1 str 9.250 | | 246 Bicarbonate of magnesia | 2 c + 1 m 8.000 | | 247 Carbonate of magnesia | 1 c + 1 m 5.250 | | 248 Carbonate of yttria | 1 c + 1 y 11.151 | | 249 Carbonate of zirconia | 1 c + 1 z 8.375 | | 250 Carbonate of glucina | 1 c + 1 g 12.584 | | 251 Carbonate of silver | 1 c + 1 s 17.500 | | 252 Percarbonate of mercury | 1 c + 2 m 56.750 | | 253 Percarbonate of copper | 1 c + 1 cop 19.750 | | 254 Carbonate of iron | 2 c + 1 i 14.615 | | 255 Carbonate of nickel | 2 c + 1 n 14.805 | | 256 Carbonate of cobalt | 2 c + 1 cob 14.826 | | 257 Carbonate of lead | 2 c + 1 l 33.500 | | 258 Carbonate of zinc | 1 c + 1 z 7.845 | | 259 Carbonate of manganese | 2 c + 1 m 13.335 | | 260 Carbonate of cerium | 2 c + 1 ce 19.349 | | 261 Percarbonate of cerium | 3 c + 1 ce 23.100 |

The bicarbonate of potash is the common crystallized carbonate of the shops. If it be a compound of 2 integrant particles of carbonic acid, and 1 integrant particle of potash, it ought to consist of 47.835 acid + 52.165 base. Now, Berthelot obtained 47.64 acid + 52.36 base, a result which may be considered as quite identical with the theory, the difference being greatly within the limits of error from experiment. Dr Wollaston has shown, that the deliquescent carbonate contains just half the carbonic acid which exists in the bicarbonate.

The supercarbonate of soda is found native in Africa, and may be formed artificially, by passing a current of carbonic acid gas through common carbonate of soda dissolved in water. According to the analysis of Klaproth, it is composed of 39 acid + 38 base. Now, the tabular statement supposes it a compound of 39 acid + 37.247 base. We have different analyses of the carbonate of soda; but they do not agree well with each other. Bergman obtained 16 acid + 20 base; Darcey 16.04 acid + 20.85 base; and Klaproth 16 acid + 22 base. The last is the most correct. According to the tabular statement, which must be correct, the salt ought to be a compound of 16 acid + 22.021 base.

The analysis of the ammoniacal salts is attended with such difficulties, that great exactness can hardly be expected, except in those cases when the acid has a gaseous form. It is then easy to determine the volume of ammoniacal gas capable of saturating a given volume of the acid gas. Carbonic acid being a gas, the composition ammoniacal carbonates may be determined with accuracy. Now, 100 measures of carbonic acid condense and neutralize 100 measures of ammoniacal gas. This will be found, on making the calculation, to be 2 integrant particles of carbonic acid, and 1 integrant particle of ammonia, constituting the first of the ammoniacal salts in the table. Fifty measures of carbonic acid are likewise capable of combining with 100 measures of ammoniacal gas, constituting the second ammoniacal salt in the table. It has not been proved that any other carbonate of ammonia exists.

Supposing carbonate of lime a compound of 1 integrant particle of carbonic acid, and 1 integrant particle of lime, it should be a compound of 43.18 acid + 56.82 lime. Now, Dr Marcet obtained 43.9 acid + 56.1 base; and Dr Thomson, by a careful analysis of very pure calcareous spar, obtained 43.2 acid + 56.8 base. This last result is almost the very same with the theoretical result.

Carbonate of barytes, by the statement in the table, is a compound of 22.04 acid + 77.96 barytes. Now, Kirwan obtained 22 acid + 78 base; and Berzelius 21.6 acid + 78.4 base. Both of these results agree sufficiently with our theoretical result; the first, indeed, coincides with it.

The analyses of carbonate of strontian hitherto made cannot be expected to agree completely with our theoretical statement, because they have been all made upon the native carbonate of that earth, which always contains a certain proportion of carbonate of lime. According to an accurate analysis by the writer of this article, it is composed of 29.9 acid + 70.1 base. Klaproth's analysis gives us 30 acid + 69.5 base. The excess of acid is obviously owing to the presence of carbonate of lime in the native salt.

We have never seen any specimen of carbonate of magnesia composed of 1 integrant particle of acid, and one integrant particle of base, in commerce. But it occurs native in abundance, constituting an ingredient in the magnesian limestone, which is so abundant in the counties of Durham, York, and Derby, and which seems to constitute a peculiar formation. The bicarbonate may be formed by passing a current of carbonic acid through water in which magnesia is suspended. The salt is soluble in water, and crystallizes. It is composed of 69.911 acid + 30.089 base. Kirwan and Fourcroy found 66.6 acid + 33.3 base. Probably their salt was mixed with a little carbonate.

Carbonate of yttria was analyzed by Klaproth; and the tabular statement exactly coincides with his result. We have no analysis of carbonates of zirconia, and glucina.

Berzelius found carbonate of silver a compound of 15.9 acid + 84.1 base. According to the statement in the table, it ought to be composed of 15.9 acid + 81.407 base, numbers which do not deviate far from those given by Berzelius.

We have no good analysis of the percarbonate of mercury. The statement in the table is given on the authority of Bergman; but is not very likely to be correct.

Percarbonate of copper, according to the analysis of Berzelius, is composed of 19.73 acid + 71.7 peroxide. Dr Thomson analyzed it, with precisely the same result. This analysis coincides exactly with the statement in the table.

Klaproth analyzed the carbonate of iron, and found it a compound of 38.3 acid + 61.6 base. This analysis corresponds sufficiently with the statement in the table.

Nickel and cobalt have so striking an analogy with iron in most particulars, that there is every reason for believing that the salts of the three metals are similarly constituted. Hence, as the carbonate of iron consists of 2 integrant particles of carbonic acid, and 1 integrant particle of black oxide of iron, we have supposed, in the table, that this is the case likewise with the carbonates of nickel and cobalt. The carbonate of nickel has been analyzed by Proust, and his result does not differ much from the tabular statement; but we have no analysis of the carbonate of cobalt.

Berzelius analyzed carbonate of lead, and found it a compound of 16.444 acid + 83.333. The analysis of other chemists corresponds nearly with this result. Now, our statement in the table supposes the salt a compound of 16.446 acid + 83.554 base, which may be considered as absolutely the same with the numbers given by Berzelius.

According to Mr Smithson, the carbonate of zinc is composed of 1 acid + 2 oxide. Now, the statement in the table supposes it a compound of 1 acid + 1.852 base; a coincidence as near as can be expected, when we consider the small quantity of salt upon which Mr Smithson makes his experiments.

If the carbonate of manganese be composed as stated in the table, it will be a compound of 34.16 acid + 50.476 base. John's analysis gives us 34.16 acid + 55.84 base; a result which does not correspond well. Hence there is reason to suspect inaccuracy.

The carbonates of cerium have been analyzed by Hisinger, and the results which he obtained agree tolerably well with the tabular statement.

Thus the carbonates correspond likewise with the Atomic Theory. They all agree with Berzelius's law, that the oxygen in the acid is a multiple by a whole number of the oxygen in the base. Most of them agree with his law, that the oxygen in the acid is double that in the base. But the carbonate of copper constitutes an exception to this rule. Indeed, we must expect to find some exceptions to these laws of Berzelius in almost every genus of salts. But it is surprising in how many unexpected cases they hold. Let us now examine the nitrates.

| Genus III.—Nitrates | Number of Atoms | Weight of an Integrant Particle | |---------------------|----------------|-------------------------------| | 262 Nitrate of potash | 1 n + 1 p | 12.750 | | 263 Nitrate of soda | 2 n + 1 s | 21.375 | | 264 Nitrate of ammonia | 1 n + 1 a | 8.875 | | 265 Nitrate of magnesia | 1 n + 1 m | 9.250 | | 266 Nitrate of lime | 1 n + 1 l | 10.375 | | 267 Nitrate of barytes | 1 n + 1 b | 16.500 | | 268 Nitrate of strontian | 1 n + 1 str | 13.250 | | 269 Nitrate of ammonia and magnesia | 4 n + 3 m + 1 a | 36.625 | | 270 Nitrate of copper | 2 n + 1 c | 23.500 | | 271 Subnitrate of copper | 1 n + 2 c | 26.750 | | 272 Nitrate of iron | 2 n + 1 i | 22.625 | | 273 Pernitrate of iron | 3 n + 1 i | 29.375 | | 274 Nitrate of zinc | 1 n + 1 l | 10.840 | | 275 Nitrate of lead | 2 n + 1 l | 41.500 | | 276 1st subnitrate of lead | 1 n + 1 l | 34.750 | | 277 2d subnitrate of lead | 2 n + 3 l | 97.500 | | 278 3d subnitrate of lead | 1 n + 3 l | 90.750 | | 279 Nitrate of nickel | 3 n + 1 nick | 29.555 | | 280 Subnitrate of nickel | 1 n + 7 nick | 71.885 | | 281 Nitrate of silver | 1 n + 1 s | 21.500 | | 282 Nitrate of mercury | 1 n + 1 m | 32.750 | | 283 Pernitrate of mercury | 1 n + 2 m | 60.750 | | 284 Subnitrate of platinum | 1 n + 4 pl | 53.250 | | 285 Nitrate of bismuth | 1 n + 1 b | 16.750 | | 286 Nitrate of uranium | 1 n + 1 u | 21.750 |

The analysis of the nitrates is attended with considerable difficulty, because we have no substance which forms an insoluble compound with nitric acid. We cannot, therefore, separate the acid by precipitation, as we do in the case of the sulphates, neither does it fly off when another acid is applied, as happens in the carbonates, so as to enable us to determine, by the loss of weight, how much acid the salt contains. Nevertheless, as a considerable number of the nitrates are destinate of water of crystallization, and composed only of the acid and base united together, we have the means of ascertaining the composition of such salts very accurately, by determining the proportion of base which they contain.

According to the analysis of Dr Thomson, nitre is composed of 100 acid + 83.823 base. According to Dr Wollaston, it is composed of 100 acid + 86.764 base. The statement in the table supposes it a compound of 100 acid + 88.8 potash. The difference between this result and experiment may be owing to our having made an atom of nitric acid a very little too great.

According to Dr Thomson, nitrate of soda is composed of 100 acid + 57.8 soda. According to Wenzel, it is composed of 100 acid + 60 base. Our tabular statement supposes 100 acid + 57.92, which almost coincides with the result of Dr Thomson.

Berzelius found nitrate of ammonia composed of 100 acid + 31.266 ammonia. Our tabular state- ment supposes the salt to be composed of 100 acid + 30.853 ammonia, which very nearly coincides with the experiment of Berzelius.

According to Wenzel, nitrate of magnesia is composed of 100 acid + 38.88 base. According to Dr Thomson, of 100 acid + 34.729 base. Our tabular statement supposes it a compound of 100 acid + 37.881 base, which comes very near the experiment of Wenzel.

According to Kirwan, nitrate of lime is composed of 100 acid + 55.70 lime. According to Dr Thomson, of 100 acid + 53.091 lime. The tabular statement supposes it composed of 100 acid + 53.2 lime; a number which comes sufficiently near experiment.

According to Berzelius, nitrate of barytes is composed of 100 acid + 140.73 barytes. The tabular statement supposes 100 acid + 143.02 base, which does not differ more than 1 per cent. from experiment.

Vauquelin informs us that nitrate of strontian is composed of 100 acid + 98.341 base. Now, our statement supposes it a compound of 100 acid + 101.42 base, which is within less than 1 per cent. of the experimental result.

We suppose nitrate of ammonia and magnesia to be a compound of 3 integrant particles of nitrate of magnesia, and 1 integrant particle of nitrate of ammonia; because that comes nearest the analysis of Fourcroy. But indeed his analyses are very seldom to be depended on.

The nitrate of copper has not been analyzed. Its composition is inferred from the sulphate of copper, which has been examined with accuracy. Berzelius found the subnitrate composed of 100 acid + 349.2 base. Proust obtained from the same salt 100 acid + 418.7 base. The mean of these two experiments gives 100 acid + 383.95 base. Now, the constitution of the salt which we have supposed in the table requires its composition to be 100 acid + 386.67 base, which comes within half a percent. of the preceding mean, and cannot, therefore, be far from the truth.

The nitrates of iron are given from theory and from analogy, as it would be scarcely possible, in the present state of the science of chemistry, to analyze them with accuracy.

Neither has the nitrate of zinc been analyzed; yet we do not think there can be any hesitation in admitting the statement given in the table to be correct.

According to Berzelius, octahedral nitrate of lead is composed of 100 acid + 205.1 oxide. This analysis, which we consider as exact, corresponds well with our statement in the table. The three subnitrates are described and analyzed by Berzelius (Annals of Philosophy, II. 278). The first contains 2 cc as much oxide as the octahedral nitrate, the second thrice as much, and the third six times as much. The second of these salts exhibits one of Berzelius's exceptions to the Atomic Theory. Its symbol would be, \(1(a^o) + \frac{1}{3}(l^o)\), or 1 integrant particle of nitric acid united with \(\frac{1}{3}\) integrant particles of oxide of lead. This appears an absurdity at first sight, because half an integrant particle can have no existence. But when stated, as we have done in the table, the absurdity vanishes: the symbol then becomes \(2(a^o) + 3(l^o)\), or two integrant particles of nitric acid, united to three integrant particles of oxide of lead. There is nothing in such an union incompatible with the Atomic Theory; though it is inconsistent with one of Berzelius's general laws, that, in all unorganic compounds, one of the elements enters always in the state of a single atom. In this compound there is no single atom. It consists of 11 atoms of oxygen, + 3 atoms of lead, + 2 atoms of azote.

Nitrate of nickel, according to Proust, is composed of 100 acid + 45.45 base. If so, its composition must be as represented in the table. Proust found the subnitrate composed of 100 acid + 735.3 oxide. This leads to the constitution of the salt in the table; if any confidence can be put in the accuracy of the analysis.

The analysis of Berzelius makes nitrate of silver a compound of 100 acid + 216.45 base. Proust gives us 100 acid + 233.33 base. The statement in the table supposes 100 acid + 216.286, which almost coincides with the result obtained by Berzelius.

Nitrate of mercury has not been analyzed. But Messrs Braamchamp and Siguiera Oliva found the pernitrate composed of 100 acid + 733.33 base. This constitutes a very near approximation to the tabular statement, especially when we consider the difficulty of the analysis.

Chenevix found the subnitrate of platinum composed of 100 acid + 809.09. This comes within \(1\frac{1}{2}\) per cent. of our statement in the table; a sufficiently near coincidence, if the difficulty of analysis be attended to.

According to Berzelius, nitrate of bismuth is composed of 100 acid + 142.69 base. Our statement supposes it composed of 100 acid + 145.42; a sufficiently near coincidence for our purpose.

Nitrate of uranium, according to Bucholz, is composed of 100 acid + 292 base. This approaches pretty nearly to the statement in the table, which supposes the salt to be composed of 100 acid + 290.4 base.

Thus, we have gone over the composition of the nitrates, without finding anything inconsistent with the Atomic Theory. On the contrary, they agree with that theory so exactly, wherever accurate methods of analysis can be had recourse to, that we may apply the theory, without hesitation, to determine the composition of those nitrates which cannot, in the present state of the science, be analyzed correctly. But, when we apply Berzelius's law, that the oxygen in the acid is a multiple of the oxygen in the base, by a whole number to the nitrates, we find that it fails in several cases, though it holds in the greatest number. As nitric acid contains 5 atoms of oxygen, it is obvious that it cannot combine with a single particle of a deutoxide, without transgressing the rule. But Berzelius has fallen upon a way of reconciling the nitrates to his law. According to him, azote is a compound of 1 atom of oxygen, and 1 atom of nitricum, so that nitric acid, according to him, is a compound of 6 atoms oxygen, and 1 atom nitricum. This opinion is certainly plausible, though it cannot be adopted till something better than mere theory is brought forward in support of it. Let us now examine the phosphates, a genus of salts still much more imperfectly analyzed than any of the three preceding genera.

**Genus IV.—Phosphates.**

| Number of Atoms | Weight of Integral Particle | |-----------------|----------------------------| | 287 Phosphate of potash | 1 ph + 1 p 9.625 | | 288 Biphosphate of potash | 2 ph + 1 p 13.250 | | 289 Subphosphate of potash | 1 ph + 2 p 15.625 | | 290 Phosphate of soda | 2 ph + 1 s 15.125 | | 291 Biphosphate of soda | 4 ph + 1 s 22.375 | | 292 Phosphate of ammonia | 5 ph + 4 a 26.625 | | 293 Biphosphate of ammonia | 5 ph + 2 a 22.875 | | 294 Quadrostophosphate of lime | 5 ph + 1 l 21.750 | | 295 Binostophosphate of lime | 5 ph + 2 l 25.375 | | 296 Bigephosphate of lime | 5 ph + 3 l 29.000 | | 297 Osteophosphate of lime | 5 ph + 4 l 32.625 | | 298 Phosphate of lime | 5 ph + 5 l 36.350 | | 299 Gephosphate of lime † | 5 ph + 6 l 39.975 | | 300 Phosphate of barytes | 1 ph + 1 b 13.375 | | 301 Phosphate of strontian | 1 ph + 1 s 10.125 | | 302 Phosphate of magnesia | 1 ph + 2 m 8.625 | | 303 Protophosphate of copper | 1 ph + 1 c 12.625 | | 304 Perphosphate of copper | 1 ph + 2 c 23.625 | | 305 Protophosphate of iron | 2 ph + 1 i 16.393 | | 306 Perphosphate of iron | 2 ph + 1 i 17.393 | | 307 Phosphate of lead | 2 ph + 1 l 35.250 | | 308 Phosphate of zinc | 1 ph + 1 z 8.720 | | 309 Biphosphate of zinc | 2 ph + 1 z 12.345 | | 310 Phosphate of nickel | 2 ph + 1 n 16.553 |

Phosphate of potash, supposing it composed of 1 integrant particle of acid, and 1 integrant particle of base, ought to consist of 100 acid + 166.762 potash. Now, according to the analysis of Saussure jun. it is composed of 100 acid + 185.714 base. According to Berthollet, it is composed of 100 acid + 199.242 base. The author of this article lately subjected all the three phosphates of potash to a careful analysis. The results agree exactly with the tabular number.

We are not acquainted with any analysis of phosphate of soda, except one made lately by Dr Thomson. He found it composed of 100 acid + 109.59 soda. Now, according to the statement in the table, it is composed of 100 acid + 109.533. In this salt, the difference between the theoretical and experimental result may be considered as nothing.

The phosphates of ammonia and of magnesia are given from analyses of these salts made by the author of this article, but not yet published.

Phosphate of magnesia and ammonium, according to the analysis of Fourcroy, is composed of equal parts of phosphate of magnesia and phosphate of ammonia. But this result not agreeing with the constituents of these salts, as determined by experiment, this triple salt has been left out of the table.

There can be no doubt that phosphate of lime is composed of an integrant particle of phosphoric acid, united with an integrant particle of lime. This is the case with the sulphate, carbonate, and nitrate of lime, and unless the phosphate of lime were so formed, Richter's law could not hold good. According to this constitution, the salt must consist of 100 acid + 100 base. We have a considerable number of analyses of phosphate of lime; but all deviating so much from each other, that little confidence can be put in any of them. According to Klaproth, it is composed of 100 acid + 254.281 lime. But the salt which he formed by saturating phosphoric acid with marble, certainly was not the common phosphate of lime, but a subsalt, which has not yet been accurately examined. According to Ekeberg (when his calculations are rectified), phosphate of lime is composed of 100 acid + 82.186 base. This approaches much nearer the true composition, though it is not correct. The salt which Ekeberg found exists, and constitutes earth of bones. It is the ostcophosphate of the table. According to Fourcroy and Vauquelin, phosphate of lime is composed of 100 acid + 143.9 lime. Here there is still a greater error on the opposite side than that fallen into by Ekeberg. But their analysis approaches to that of apatite, the gephosphate of the table. The other phosphates of lime in the table were discovered by the author of this article, and are detailed by him in a paper on phosphoric acid, read to the Royal Society, but not yet published.

Phosphate of barytes, according to the experiments of Berzelius, is composed of 100 acid + 259.715 barytes. Our statement in the table, founded on evidence, which seems quite satisfactory, supposes it a compound of 100 acid + 270.477 base. The difference here between experiment and theory does not exceed 3 per cent.

If phosphate of strontian be composed of 1 integrant particle of phosphoric acid, and 1 integrant particle of strontian, as there can be no doubt is the case, then its composition must be 100 acid + 191.773 base. We have an analysis of this salt by Vauquelin, according to whom it is composed of 100 acid + 192.48 base. But the data upon which he founded his conclusions, were necessarily so inaccurate, in consequence of the erroneous notions which he entertained respecting the composition of phosphate of lime, that a nearer approach to accuracy could not, in his case, be expected. The numbers in the table are the result of experiments by the author of this article.

It seems to be a law in saline combinations, that a deutoxide always combines with two integrant particles of acid. Thus, black oxide of copper unites with two integrant particles of sulphuric acid, and likewise with two integrant particles of nitric acid.

---

* So called, because it is a constituent of bones. † This is the apatite of mineralogists. Hence we may conclude, that it will combine likewise with two integrant particles of phosphoric acid. According to this statement, it must be composed of 100 acid + 138.854 black oxide of copper. Now, according to the analysis of Chenevix, it is composed of 100 acid + 141.428 black oxide of copper. This agrees with the tabular statement very nearly.

We have the same reason for considering phosphate of iron as a compound of two integrant particles of acid, and one of deutoxide of iron. According to this statement, it ought to be composed of 100 acid + 126.667 black oxide of iron. We have no analysis of this sort. Lougier, indeed, has analyzed a substance which occurs in small prisms in Brazil, and in the Isle of France. He found it composed of 100 acid + 214.285 black oxide. This approaches to a compound of 1 integrant particle of acid, and 1 integrant particle of oxide. Dr Thomson found two phosphates composed as in the table.

The perphosphate of iron is a white insoluble powder, obtained by precipitating persulphate of iron by phosphate of ammonia. It has not been analyzed. Nor do we know whether the theoretical result, as stated in the table, can be admitted to much confidence. This salt is converted into a subsalt, by the action of an alkali.

Phosphate of lead has been analyzed with great care by three different chemists. According to Berzelius, it is composed of 100 acid + 380.565 base; according to Dr Wollaston, of 100 acid + 370.72 base; and according to Dr Thomson, of 100 acid + 398.49 base. The mean of these experiments gives us 100 acid + 38, nearly 3.258 base. This agrees with the composition of the salt, as stated in the table. It was from it, in part, that we deduced the weight and composition of phosphoric acid.

The phosphates of zinc, and phosphate of nickel, have been analyzed by Dr Thomson. The composition of these salts in the table is stated from this analysis.

There is no analysis of the phosphate of mercury. We do not even know if it exists. But the perphosphate of mercury has been analyzed by Messrs Braamcamp and Siguiera. According to them, it is composed of 100 acid + 250.857 peroxide of mercury. Now, if it be a compound of 3 integrant particles of acid and 1 integrant particle of base, its constituents ought to be 100 acid + 250.159 base. This may be considered as identical with the experiment, since the difference is far within the limits of unavoidable error, from the imperfection of our methods.

Thus it appears, not only that no exception to the atomic theory exists among the salts; but that it puts it in our power to determine the composition of all the genera of salts, provided we know the analysis of one species, with more accuracy than would result from ordinary experiment. We have not the least doubt, that, when an accurate analysis of the phosphates is published, it will be found in almost every instance to agree with our table. We shall give another example of a genus of salts, the constituents of which we have determined almost entire-

**Genus V.—Borates.**

| Number of Atoms | Weight of an Integrant Particle | |-----------------|--------------------------------| | 311 Borate of potash | 1 b + 1 p | 8.733 | | 312 Sub-borate of potash | 1 b + 2 p | 14.733 | | 313 Borate of soda | 2 b + 1 s | 13.341 | | 314 Sub-borate of soda | 1 b + 2 s | 18.483 | | 315 Borate of ammonia | 1 b + 1 a | 4.858 | | 316 Sub-borate of ammonia | 1 b + 2 a | 6.983 | | 317 Borate of magnesia | 1 b + 1 m | 5.283 | | 318 Borate of lime | 1 b + 1 l | 6.358 | | 319 Borate of barytes | 1 b + 1 bar | 12.483 | | 320 Borate of strontian | 1 b + 1 str | 9.233 | | 321 Borate of alumina | 1 b + 1 a | 4.858 | | 322 Borate of yttria | 1 b + 1 y | 11.138 | | 323 Borate of glucina | 1 b + 1 g | 12.566 | | 324 Borate of zirconia | 1 b + 1 z | 8.358 | | 325 Borate of copper | 2 b + 1 c | 15.466 | | 326 Borate of iron | 2 b + 1 i | 14.581 | | 327 Borate of nickel | 2 b + 1 n | 14.771 | | 328 Borate of cobalt | 2 b + 1 c | 14.792 | | 329 Borate of lead | 2 b + 1 l | 33.466 | | 330 Borate of zinc | 1 b + 1 z | 7.828 | | 331 Borate of mercury | 1 b + 1 m | 28.733 | | 332 Borate of silver | 1 b + 1 s | 17.483 | | 333 Borate of bismuth | 1 b + 1 b | 12.733 | | 334 Borate of manganese | 2 b + 1 m | 10.566 | | 335 Borate of uranium | 1 b + 1 u | 15.733 | | 336 Borate of platinum | 1 b + 1 p | 15.894 |

We shall take one other genus of salts, as a farther elucidation of the atomic theory; and we shall select one of the vegetable acids, which have been analyzed by Berzelius, that we may be able to judge how far the weight of an integrant particle of the acid derived from his analysis, will agree with the constitution of the salts.

**Genus VI.—Oxalates.**

| Number of Atoms | Weight of an Integrant Particle | |-----------------|--------------------------------| | 337 Oxalate of potash | 1 ox + 1 p | 10.634 | | 338 Binoxalate of potash | 2 ox + 1 p | 15.268 | | 339 Quadroxalate of potash | 4 ox + 1 p | 24.536 | | 340 Oxalate of soda | 2 ox + 1 s | 17.150 | | 341 Oxalate of ammonia | 1 ox + 1 a | 6.783 | | 342 Oxalate of magnesia | 1 ox + 1 m | 7.211 | | 343 Oxalate of lime | 1 ox + 1 l | 8.254 | | 344 Binoxalate of lime | 2 ox + 1 l | 12.888 | | 345 Oxalate of barytes | 1 ox + 1 b | 14.365 | | 346 Oxalate of strontian | 1 ox + 1 str | 11.534 | | 347 Oxalate of alumina | 1 ox + 1 a | 6.770 | | 348 Oxalate of yttria | 1 ox + 1 y | 13.034 | | 349 Oxalate of glucina | 1 ox + 1 gl | 14.467 | | 350 Oxalate of zirconia | 1 ox + 1 z | 10.290 | | 351 Oxalate of copper | 2 ox + 1 c | 19.268 |

VOL. I. PART II. According to the analysis of Berzelius, exhibited in a preceding part of this article, oxalic acid is a compound of 21 atoms, namely, 12 atoms of oxygen, 8 atoms of carbon, and 1 atom of hydrogen. But there can be little hesitation in doubling the quantity of hydrogen found in this acid by Berzelius; because this will greatly increase the simplicity of the composition of the acid, and the difference is considerably within the unavoidable errors in such experiments. In that case, the acid would be constituted of 11 atoms; namely, 6 of oxygen, 4 of carbon, and 1 of hydrogen; and its weight would be 9.070. But the weight of an atom of oxalic acid, even when thus corrected, will not agree with the known constitution of the oxalates. Berzelius analyzed oxalate of lead with great care, and found it composed of 100 acid + 307.5 oxide of lead. Now, it appears from all the genera of salts which we have examined, that the yellow oxide of lead unites always with 2 integrant particles of acid. The weight of an integrant particle of yellow oxide of lead is 28.

Now, \( \frac{307.5}{2} : \frac{100}{2} : 57 : 4.528 = \) the weight of an integrant particle of oxalic acid. Now, this is very nearly one half of the weight resulting from analysis. To obtain this weight, it would be necessary to quadruple the quantity of hydrogen found in oxalic acid by Berzelius, and to consider it as composed of 3 atoms oxygen, 2 atoms carbon, and 1 atom hydrogen. Now, it deserves to be noticed, that this is the constitution of oxalic acid adopted by Mr Dalton (Annals of Philosophy, III. 179.), obtained, we presume, from an examination of the composition of the oxalates, and from the analysis of oxalic acid, previously made by Dr Thomson and by Gay-Lussac and Thenard. Whatever confidence the analytical experiments of Berzelius may be entitled to, we have no alternative in the present case but to adopt the conclusion suggested by the oxalates. The analysis of these bodies is much easier, and therefore more likely to prove correct, than that of oxalic acid itself. We may adopt 4.625 as the weight of an atom of oxalic acid; and consider it as a compound of 6 atoms, 3 of oxygen, 2 of carbon, and 1 of hydrogen. Its composition would then be:

\[ \begin{align*} \text{Oxygen} & : 64.789 \\ \text{Carbon} & : 32.413 \\ \text{Hydrogen} & : 2.848 \\ \end{align*} \]

We do not pretend to reconcile this result with that obtained by Berzelius; but the constitution of the oxalates obliges us to adopt it as nearest the truth.

We have three analyses of the oxalate of potash, one by Dr Thomson (Phil. Trans. 1807), one by Vogel (Schweigger's Journal, II. 470), and one by M. Berard (Annales de Chimie, 73. 270). The statement in the table supposes the salt a compound of 100 acid + 129.477 potash. Now, Dr Thomson's analysis gives us 100 acid + 122.86 base, Vogel's 100 acid + 132.558 base, and Berard's 100 acid + 102.757 base. The first two analyses, especially the second, are very near the truth. Berard's is a good deal more incorrect. Dr Wollaston has shown, that binoxalate of potash contains exactly twice as much acid, and quadroxalate of potash four times as much acid as neutral oxalate of potash.

If oxalate of soda be composed of two integrant particles of acid, and 1 of soda, it must be composed of 100 acid + 85.045 soda. Now, the analysis of Vogel gives us 100 acid + 82.563 base, which agrees very nearly with theory. The analysis of Thomson and Berard are both considerably more incorrect.

Supposing oxalate of ammonia to be constituted as represented in the table, it must be a compound of 100 acid + 46.374 ammonia. Now, the analysis of Berzelius gives us 100 acid + 45.264 base, and that of Berard 100 acid + 44.369 base. Both of these agree well with the tabular statement.

Superoxalates of soda and of ammonia exist, but we did not give them a place in the table, as they have been but superficially examined. The first is probably a compound of 3 integrant particles of acid, and 1 integrant particle of soda; the second no doubt contains 2 integrant particles of acid and 1 of ammonia. Oxalate of magnesia, supposing it constituted as in the table, is a compound of 100 acid + 55.610 magnesia. Now, Bergman's analysis gives us 100 acid + 53.846 base. This agrees sufficiently with the theory. The analysis of Thomson and Berard are both very faulty. It is not difficult to see how they were misled.

Oxalate of lime has been analyzed with great care by different chemists, Bergman, Thomson, Berard, Vogel, Gay-Lussac, and Thenard, &c. But the results differ considerably from each other. The reason is, that various oxalates of lime exist, and that experimenters are always sure to get a mixture of them, when they form oxalate of lime by precipitation. The neutral salt must be composed of 100 acid + 78.117 lime. Now, Vogel's analysis gives us 100 acid + 76 lime, which is sufficiently near to show us, that he calculated the composition of the neutral salt. Dr Thomson gives 100 acid + 60 lime, this must have been a mixture of neutral oxalate and binoxalate of lime. Berard found 100 acid + 61.2 lime, and Thenard, and Gay-Lussac, nearly the same proportions. All these must have been mixtures.

The analyses of the oxalate of barytes hitherto made, do not correspond with its composition as represented in the table; owing, probably, to the same causes that have occasioned inaccuracy in the analysis of oxalate of lime. This salt ought to be composed of 100 acid + 209.991 barytes. Dr Thomson's analysis gives us 100 acid + 142.86 base, and Berard's 100 acid + 164.3 base. Both of these are very far from accurate.

Oxalate of strontian, according to the statement in the table, ought to be composed of 100 acid + 148.894 strontian. Now, according to Dr Thomson's analysis, it is composed of 100 acid + 151.51 base. This agrees sufficiently. Berard's analysis is much less accurate. He obtained 100 acid + 119.5 base. The oxalates of alumina, yttria, glucina, and zirconia, have not been analyzed. But there can be no hesitation in considering their composition as stated in the table to be correct.

Oxalate of copper, according to the constitution of it given in the table, is composed of 100 acid + 107.898 black oxide of copper. Now, Vogel found it composed of 100 oxide + 100 acid, including a little water. This agrees sufficiently with the theoretic statement. The three triple oxalates are composed respectively as follows:

1. One integrant particle oxalate of potash, + 1 integrant particle oxalate of copper. 2. 1 oxalate of soda + 1 oxalate of copper. 3. 1 oxalate of ammonia + 1 oxalate of copper.

This agrees with Vogel's analysis. He found oxalate of potash and copper composed of

| Acid | Potash | Oxide of copper | |------|--------|----------------| | 45 | 30 | 25 |

100

And oxalate of ammonia and copper of

| Acid | Ammonia | Oxide of copper | |------|---------|----------------| | 47.5 | 10.5 | 25.0 |

100.0

The remaining oxalates, if we except the oxalate of lead, have not yet been subjected to an accurate analysis. The composition, as stated in the table, is deduced from a comparison with the other genera of salts, and there can be no doubt that, in general, it is correct.

It would be easy for us to exhibit the composition of the remaining genera of salts; but we have already extended this article to as great a length as is consistent with our limits. The remaining genera, indeed, have been so imperfectly examined, that they would throw no additional light on the atomic theory; though that theory would enable us to calculate their compositions in most cases with sufficient precision. We have omitted, likewise, the chlorides and iodes, two numerous classes of bodies analogous to the oxides to which the atomic theory applies, with as much precision as it does to those substances which we have examined in this article. But these two classes of bodies being but recently discovered, are not probably so familiar to the reader, as to warrant our introducing them without any previous explanation.

(J.)

**Attraction.**

The word **Attraction** (see *Encyclopaedia*), is used to denote what we observe, when one body approaches another, or tends to approach it, without any apparent impulse, or other cause, to which the motion can be ascribed.

We have instances of attraction when iron approaches the magnet,—when certain bodies are placed near an excited electric,—when a stone falls to the earth. We say likewise that the earth attracts the moon; by this mode of expression meaning no more than that the moon is continually deflected towards the earth, from the rectilineal course which it would otherwise pursue. It is likewise in this sense that we must be understood, when we say that the sun attracts all the planets.

In the instances already mentioned, attraction extends to a distance. In other cases, it is confined within limits so extremely narrow, as to become imperceptible at an interval which cannot be appreciated by the senses. Of this kind is the attraction which takes place between the particles of the same fluid, as is apparent from the round figure of small drops. To this class likewise belongs the attraction between fluid and solid bodies; whence originate the very interesting appearances observed in capillary tubes, and other kindred phenomena. An attraction between the small elementary particles of all solid bodies, is manifest from the force with which they cohere, or resist an endeavour to separate them. In many cases, the intensity of this force is prodigiously great in contact, or at the nearest distances; while it ceases to act, upon making the smallest separation between the parts. Lastly, Chemistry develops innumerable instances of attraction between the molecules of the bodies about which it is conversant; insomuch, that it is to this principle, under the name of affinity, Attraction. that we must ultimately ascribe the various decompositions and new combinations which occur in that science.

All these phenomena, although very different from one another in other respects, yet have this in common, that we observe, in certain bodies, a tendency to approach one another, and to resist a separation, with some degree of force. The facts are certain, and are attended with no ambiguity; and it is to express these facts that the term attraction is used in physics.

We likewise observe, in some bodies, a tendency to fly off from one another when they are brought near. This is called repulsion.

The word force has, in general, some degree of obscurity. It is used to denote the cause of motion; but we have no direct knowledge of it, and we judge of its intensity by the effect which we suppose it to produce. In all our reasoning concerning forces, it is the changes of motion which we measure and compare together, and which are really the subjects of our thoughts. Attraction and repulsion are forces, or principles of motion, known to us only by the phenomena we observe; but the circumstance of their implying action at a distance, is an additional source of obscurity, in which other kinds of force do not participate.

It certainly is inconceivable, that motion should be produced at a distance, when no connection can be traced between the body moved and that which is supposed to produce the motion. We are strongly impressed with the prejudice, that a body cannot act but where it is; and we find difficulty in admitting that the mere presence of two bodies, without the intervention of any mechanical means, can be a satisfactory cause of motion. On this account, Attraction has been classed by some with the occult qualities of the schools; and the favourers of this doctrine have been reproached with reviving exploded notions in philosophy. Impulse is a principle of motion more familiar to us, and to which we are not disposed to make equal objection. Whenever the communication of motion can be traced to this source, we are satisfied that the effect is justly explained. Hence, many philosophers have been of opinion, that impulse is the only cause of motion that can be admitted in physical science; and many attempts have been made to reduce to this principle, all cases in which distant bodies act on one another.

With regard to these attempts, it will be sufficient to remark here, that they are all built upon hypothesis: no evidence is adduced to prove that such things exist, as the elastic ether, or gravific matter, which they set out with supposing. And, as far as such systems have no other object but to obviate the difficulty of action at a distance, this argument alone is sufficient to confute them, without adverting to the difficulties that attend each of them separately—their inconsistency with the received laws of motion—and the innumerable contradictions and improbabilities to which they are liable on every side.

A little reflection is sufficient to show, that, in reality, we have no clearer notion of impulse as the cause of motion, than we have of attraction. We can as little give a satisfactory reason why motion should pass out of one body into another, on their Attraction contact, as we can, why one body should begin to move, or have its motion increased, when it is placed near another body. It is equally impossible in both cases to prove that there is a necessary connection between the related facts; in this respect, both the phenomena are alike inexplicable.

When motion is produced by impulse, it is probably the circumstance of contact apparently taking place, which leads us to think that the effect is so clearly explained. It is in this manner only, or by actual contact, that we ourselves can move external objects. We have no power to produce motion in distant bodies, except by the intervention of other bodies on which we act immediately. Impulse is, therefore, a cause of motion familiar to us, and strikes us as the plainest, and most satisfactory, ultimate principle at which we can arrive. On the other hand, when one body attracts another at a distance, there is nothing familiar to us with which we can compare it; our curiosity is excited, and we are led to seek out some hidden connection between them.

But, it may be doubted whether there is actual contact in any case of the communication of motion. When a body is impelled by the air, it will hardly be affirmed that the particles of that elastic fluid are in contact with one another, since there is no space, however small, within which a given bulk of it may not be compressed, by applying an external force sufficiently great. The particles of air, therefore, act on one another at a distance; and the same thing must be true of all other elastic fluids. And, by the way, what is here said is sufficient to prove, that no scheme, founded on the hypothesis of an elastic ether, will enable us to account for attraction; because such a contrivance can do nothing more than substitute one species of action at a distance in the room of another. There is good reason to think that absolute contact never takes place in the component parts of the hardest and most compact solid bodies. This seems to be an unavoidable consequence of the fact, well established by experience, that all bodies contract in their bulk by cold, and expand by heat. It is, therefore, not only not impossible, but it is even in some degree probable, that the communication of motion may, in every instance, be a case of action at a distance.

If, then, we are apt to think that impulse is a clearer physical principle than attraction, there is, in reality, no good ground for the distinction; it has its origin in prejudice, and in our mistaking the proper object of natural philosophy. All our researches in nature are confined to the phenomena we observe, and to the laws by which they are regulated. A physical cause is no other than a general fact discovered by a careful observation, and an attentive comparison, of many particular and subordinate facts. We have no evidence, independent of experience, that any consequence, deduced from a physical cause, will actually take place. There is, in this case, no necessary connection from which we can, with absolute certainty, infer the expected event. If, then, we regard impulse and attraction as princi- Attraction. ples founded in fact, and regulated by laws, confirmed by observation and experiment, they are both equally entitled to be classed as physical causes, and they ought both to be admitted as of equal authority in explaining the phenomena of the universe.

If we turn our attention to the different kinds of attraction enumerated above, and inquire what progress has been made in the investigation of their laws of action, we shall find that, generally speaking, this branch of physics is little advanced. We are very imperfectly acquainted with magnetical and electrical attraction. We know still less of those attractive powers which take place at small distances, and which are confined within such narrow limits, that their mode of action escapes the observation of our senses. Attraction is, indeed, much used by philosophers to account for many important natural phenomena; but their explanations are often vague, and destitute of that precision which ought always to be aimed at in physical science. There is only one class of phenomena in which the laws of attraction have been fully developed. We allude to gravitation, that principle which occasions the fall of heavy bodies at the surface of the earth, and which retains the planets and comets in their orbits. Referring the other species of attraction, which are little susceptible of general discussion, to their several heads, we shall now confine our attention to gravitation.

Traces of the principle of gravitation are to be found in writers of great antiquity. But their speculations on this subject do not go beyond a vague notion of a tendency which the planets have to one another, or to a common centre. It would contribute little either to entertainment or instruction, to collect all the passages of ancient authors that speak of this principle. The revival of the true system of the world by Copernicus, introduced the most admirable simplicity in the explanation of the planetary motions, and likewise led to more just conjectures concerning the laws by which they are upheld. Copernicus himself attributed the round figure of the planets to a tendency which their parts possess of uniting with one another; thus extending to all the planets which we observe at the surface of the earth. He stopt short, indeed, at this point; conceiving attraction to be confined to the matter of each planet, without making it extend from one planet to another, so as to actuate all the bodies of the system. This step was made by the bold and systematic genius of Kepler. Adopting the opinion of Dr Gilbert of Colchester, that the earth is a great magnet, Kepler formed to himself a notion of attraction, in some respects remarkably just. He says that the earth and moon attract one another; and, were it not for some powers which retain them in their orbits, they would move towards one another, and would meet in their common centre of gravity. He attributes the tide to the moon's attraction (virtus tractive qua in luna est), which heaps up the waters of the ocean immediately under her. But in many respects, his notions of attraction were fanciful and extravagant; a more perfect knowledge of the laws of motion than had been attained to in his time, and a new geometry, were both wanting, in order to guide him in this research without danger of wandering. Yet he was able to penetrate so far into the causes of the planetary motions, as to foresee that they would not long continue latent; he tells us, he was persuaded that "the full discovery of those mysteries was reserved for the next age, when God would reveal them." So full an exposition of a physical system of the world, as is contained in the writings of Kepler, could not fail to draw the attention of succeeding philosophers. Many remarks concerning the principle of gravitation are to be found in the writings of Fermat, Roberval, Borelli, and other authors. But no one, before Newton, entertained so clear and systematic a view of the doctrine of universal gravitation as Dr Robert Hook. In his work on the motion of the earth, published in 1674, twelve years before the appearance of Newton's Principia, he lays down these three positions as the foundations of his system, viz.

"1st, That all the heavenly bodies have not only a gravitation of their parts to their own proper centre, but likewise that they mutually attract each other within their spheres of action.

"2ndly, That all bodies having a simple motion, will continue to move in a straight line, unless continually deflected from it by some extraneous force, causing them to describe a circle, or an ellipse, or some other curve.

"3rdly, This attraction is so much the greater, as the bodies are nearer."

The principle of universal gravitation is here very precisely enunciated. Dr Hook seems to have clearly perceived, that the planetary motions are the result of an attraction towards the sun, and of a rectilineal motion produced by a projectile force. Not having discovered the law according to which the force diminishes, as the distance from the sun increases, he contrived experiments to elucidate his theory. Having suspended a ball by means of a long thread, he placed another ball upon a table immediately under the point of suspension, and he caused the suspended ball to revolve round the stationary one. When the moveable ball was pushed laterally with a force properly adjusted to its deviation from the perpendicular, it described an exact circle round the ball on the table; in other cases, it described an ellipse, or an oval resembling an ellipse, having the other ball in the centre. Dr Hook observed, that although this experiment, in some measure, illustrated the planetary motions, yet it did not represent them accurately; because the ellipses which the planets describe, have the sun placed in one focus, and not in the centre. Thus, at the appearance of Newton, many things were known, or rather surmised, that prepared the way for the discovery of the principle which regulates the celestial motions. This does not detract, in any degree, from the glory of Newton, who, discarding the conjectures of his predecessors, proposed to himself to investigate, with mathematical strictness, the law of the attractive force, and to ascertain, with precision, its sufficiency to retain the planets in their orbits. He invented a new kind of geometry, which was necessary to enable him to accomplish his purpose. With this help, and by admitting no- Attraction.

thing without the sanction of the established principles of Dynamics, he deduced from the motions of the celestial bodies, the law of universal gravitation, the most important and the most general truth hitherto discovered by the industry and sagacity of man, viz. "That all the particles of matter attract one another, directly as their masses, and inversely as the squares of their distances." The particular occasion which gave rise to the speculations of Newton, on the subject of gravity, is noticed in the life of that great man in the Encyclopedia; and, in the article Astronomy (Part IV. Chapter II.) there is an account of the several steps of the analytical process of reasoning, by which the above general law is inferred from the motions of the celestial bodies.

Having arrived at a principle which belongs to every part of matter, another inquiry comes into view. Setting out from this principle, it is now necessary to proceed in an inverted order, and deduce from it, by synthetical reasoning, the phenomena which we observe in the universe. The first step in this process, is to find out the attractive force of the planets, which arises from the united attractions of their component parts. Two things only are involved in this investigation, viz. the known law of attraction between the particles of matter, and the figure of the attracting bodies. This is a subject of great importance, and it is connected with some principal points of the system of the world, with the theory of the figure of the planets, that of the tides, and many other phenomena. It is but imperfectly discussed in Newton's immortal work; and there is no part of his philosophy which has been improved more slowly by the labours of his followers. We now propose to treat of it at some length, endeavouring to lay before our readers as complete a view of this part of science as the nature of our work will permit.

We begin with laying down some definitions, and demonstrating some properties of elliptical spheroids.

Def. 1. A solid generated by the revolving of an ellipse about either axis is called a spheroid of revolution. If the ellipse revolve about the less axis, the spheroid is oblate; if about the greater axis, it is oblong.

Let \( k \) and \( k' \) denote the two axes of the spheroid, \( k \) being that of revolution; and let \( x \) and \( y \) be two coordinates of a point in the surface of the spheroid, having their origin in the centre, \( x \) being parallel to the axis of revolution, and \( y \) perpendicular to it; then the equation of the spheroid, whether oblate or oblong, will be

\[ \frac{x^2}{k^2} + \frac{y^2}{k'^2} = 1. \]

Def. 2. An elliptical spheroid, in general, or an ellipsoid, is a solid bounded by a finite surface of the second order. Let ACB and ADE (Plate XXX. fig. 1). This figure represents one eighth of an ellipsoid contained in one of the solid angles formed by the three principal sections) be two ellipses that have the same axis AO, the same centre O, and their planes perpendicular to one another; from any point K in the common axis, let there be drawn ordinates in both ellipses, as KC and KD; then, having described an ellipse of which KC and Attraction. KD are the semiaxes, the periphery DMC of that ellipse will be in the surface of the ellipsoid. This solid figure has a centre; three axes crossing one another at right angles in the centre; and three principal sections made by planes passing through every two of its axes.

Let \( k, k', k'' \), denote the three semiaxes; viz. \( k = OB, k' = OE, k'' = OA \); and let \( x, y, z \) denote three rectangular coordinates of a point M in the surface, the coordinates being parallel to the axes, and having their origin in the centre; viz. \( MN = x, NK = y, OK = z \); then the equation of the surface will be

\[ \frac{x^2}{k^2} + \frac{y^2}{k'^2} + \frac{z^2}{k''^2} = 1; \]

it is easy to prove from the foregoing construction.

The ellipsoid becomes a sphere, when all the three axes are equal; it becomes a spheroid of revolution, when two of them are equal.

1. If any plane cut an elliptical spheroid, the section will be an ellipse. In the spheroid of revolution, a section made by a plane perpendicular to the axis of revolution is a circle.—All this follows so easily from the nature of the solids, that we need not stop to give a formal demonstration.

2. If a straight line cut two concentric ellipses, that are similar and similarly situated, the parts of it between the outer and inner peripheries are equal to one another.

Let AHBK and MDNC (Plate XXX. fig. 2.) be two similar and similarly situated ellipses that have the same centre O; and let the straight line AB cut them both; then AC and BD are equal. Bisect CD in L, and through L and the common centre draw the straight line HMNK to cut both ellipses. Because the ellipses are similar and similarly situated, and that CD is an ordinate of the diameter MN, it is plain that AB will be an ordinate of the diameter HK; therefore, AB and CD being both bisected in L, AC is equal to BD.

3. If there be two ellipses, one within the other, such that, any straight line being drawn to cut them, the parts of it between their peripheries are equal to one another; these ellipses are concentric, similar, and similarly situated.

Let D (fig. 2.) be any point in the inner ellipse, and through D draw EF, terminating in the outer ellipse; then, if we make FG = DE, G must be a point in the inner ellipse. Hence all the points of the inner curve are determined, when the outer ellipse and the point D are given; wherefore there cannot be two different curves, both passing through D, that will answer the conditions. But an ellipse described through D, concentric with the outer ellipse, and similar to it, and similarly situated, will answer the conditions (2). Wherefore the two ellipses are concentric, and similar, and similarly situated.

4. If a straight line be drawn to cut two elliptical spheroids, that have the same centre, and are similar and similarly situated, the part of it between the outer and inner surfaces will be equal to one another.

Conceive a plane, which contains the straight line, to pass through the common centre of the solids; Attraction. the sections made by the plane will be concentric ellipses (1.) ; and these will be similar and similarly situated, because the solids are so; wherefore the parts of the straight line between the surfaces are equal (2.).

5. If two elliptical spheroids that have the same centre, and are similar and similarly situated, be cut by a plane, the two sections will be concentric ellipses that are similar and similarly situated.

For the sections are ellipses (1.) ; and, any straight line being drawn to cut them, the parts of it between the peripheries will be equal (4.) ; wherefore the ellipses are concentric, similar, and similarly situated (3.)

6. Let ADE and CFG (fig. 3.) be two concentric ellipses that are similar and similarly situated; let AO and CO, in the same straight line, be two of their axes, and let DE, drawn through C, be perpendicular to AO; then if CF and CG be two chords of the interior ellipse that make equal angles with the axis CO, and if the chords DM and DN of the exterior ellipse be drawn respectively parallel to CF and CG; the sum of CF and CG will be equal to the sum or difference of DM and DN, according as they both fall on the same side, or on different sides of DE.

For draw EP parallel to CF, and it will likewise be parallel to DM. Because CF and CG are equally inclined to CO and to DE, it is plain that DN and RP, which are parallel to CF and CG, are likewise equally inclined to DE; consequently DN = EP.

Draw a straight line through the common centre to bisect DM in L, and that straight line will likewise bisect EP, parallel to DM, in H; and because the ellipses are similar and similarly situated, the same straight line will likewise bisect the chord CF of the interior ellipse, in K. Because DC = CE, therefore DL + EH = 2CK = CF. Wherefore DM + DN = 2DL + 2EH = 2CF = CF + CG.

The demonstration of the other case, when DM and DN fall on different sides of DE, is entirely similar.

Some general Properties, resulting from the Law of Attraction, that obtains in Nature.

7. Let AB and EF (fig. 4.) be two indefinitely slender pyramids, that are similar to one another, and both composed of the same homogeneous matter, which attracts in the inverse proportion of the square of the distance; the attractions of the pyramids upon particles placed at the vertices A and E, are proportional to the length of the pyramids.

Conceive each of the pyramids to be divided into an indefinitely great number of thin slices of equal thickness, by planes parallel to its base; then, if CD and GH be any two of these slices, their attractions upon particles placed at A and E, will be proportional to \(\frac{CD}{AC}\) and \(\frac{GH}{EG^2}\). Now, these are equal: for, the solids CD and GH, having the same thickness, they are proportional to the sections CM and GN, that is, to \(AC^2\) and \(EG^2\), because the pyramids are similar. Wherefore, the attraction of any one of the slices in the pyramid AB, upon a particle placed at A, is equal to the attraction of any one of the slices in EF upon a particle placed at E. Consequently, the whole attraction of the pyramid AB, is to the whole attraction of the pyramid EF, as the number of slices in AB to the number of slices in EF, that is, as the length AB to the length EF.

Cor. 1. The attractions of any portions of the pyramid, are as the lengths of the portions. For the attractions are proportional to the number of slices in the portions, that is, as the lengths.

Cor. 2. If the pyramids have different densities, their attractions are proportional to the lengths multiplied by the densities. For, in this case, the attraction of each slice will be proportional to its density: wherefore, the attractions will be as the densities multiplied by the number of slices; or, as the densities multiplied by the lengths.

8. If there be two similar solids composed of the same homogeneous matter, which attracts in the inverse proportion of the square of the distance; any two particles of matter, similarly situated with regard to the solids, will be attracted by them with forces that are proportional to any of the homologous lines of the solids.

Because the solids are similar, they may be resolved into an indefinitely great number of slender pyramids, and frustums of pyramids, that are similar to one another, and similarly placed in the solids; each pyramid having its vertex at one of the attracted particles. The direct attractions of any corresponding pair of pyramids will have constantly the same ratio to one another: for they will be as the lengths of the pyramids or frustums (7.) ; that is, because the solids are similar, as any two homologous lines of the solids. Therefore, the whole attractive forces, compounded of all the direct attractions which act in directions that make the same angles with one another, will likewise have to one another the proportion of any two of the homologous lines of the solids.

Cor. If the two solids have different densities, their attractions will be proportional to the densities multiplied by any homologous lines of the solids (7. Cor. 2.).

9. If there be two concentric elliptical spheroids that are similar and similarly situated, a particle placed anywhere within the inner surface will be in equilibrium, or will be urged equally in all opposite directions by the shell of homogeneous matter contained between the two surfaces, supposing the law of attraction to be that of the inverse proportion of the square of the distance.

Let P (Plate XXX. fig. 5.) be a particle placed within such a shell, and let a slender double pyramid, having P for the common vertex, be extended to meet the surfaces of the solid on both sides of P. The portions of the pyramid, AGHB and CEF between the surfaces on opposite sides of P, will have equal lengths (4.) ; wherefore, these portions will attract a particle placed at P with equal forces (7. Cor. 1.). The same thing may be proved of all the pyramids which have their vertices at P, and fill the spheroids. Wherefore, P is attracted equally in all opposite directions by the homogeneous matter contained between the surfaces of the spheroids.

10. To find the attractive force of an indefinitely slender prism, acting in a direction parallel to the prism, upon a particle of matter placed anywhere. Let BC (Plate XXX. fig. 6.) be a prism of homogeneous matter, upon the indefinitely slender base CH, and let a particle of matter be placed at A; draw AB and AC to the extremities of the prism, and AE to any point in it; and draw AD perpendicular to BC. Let S = base CH, and put AD = a, DE = x; the element of the prism is \( S \times dx \); the element of the attraction in the direction AE, is \( \frac{S \times dx}{AE^2} \); and the element of the attraction in the direction parallel to the prism, is \( \frac{S \times dx}{AE^2} \times \frac{DE}{AE} = \frac{S \times dx}{(a^2 + x^2)^{\frac{3}{2}}} \).

Now, \( \int \frac{S \times dx}{(a^2 + x^2)^{\frac{3}{2}}} = \text{Const.} - \frac{S}{\sqrt{a^2 + x^2}} = \text{Const.} \)

\( \frac{S}{AE} \); and the constant quantity is determined by making the fluent begin at the end of the prism nearer to A; wherefore, the whole attractive force of the prism, in the direction parallel to the prism, is

\[ S \times \left\{ \frac{1}{AB} - \frac{1}{AC} \right\}. \]

Cor. In like manner may the attractive force of the prism be found, when the attraction of the particles is proportional to any function of the distance.

Let \( AB = f, AC = f' \); suppose that \( \varphi(f) \) is the function of the distance that expresses the law of attraction; and put \( \int df \cdot \varphi(f) = \Psi(f) \); then the attraction parallel to the prism, is

\[ S \times \left\{ \Psi(f) - \Psi(f') \right\}, \]

observing that the attraction is always positive.

Attraction of Spheres.

11. Spheres of the same homogeneous matter, attract particles placed on their surfaces, with forces proportional to their radii.

Spheres being similar solid figures, this proposition is no more than a particular case of what was before proved (8.).

Cor. If the spheres have different densities, the attractions at their surfaces are proportional to their radii multiplied by their densities (8. Cor.).

12. The force with which a particle, placed anywhere within a sphere of homogeneous matter, is urged towards the centre, is proportional to its distance from the centre.

Conceive a concentric sphere to be described, which contains the attracted particle in its surface; the matter between the two surfaces will exert no force on the particle (9.), which will therefore be urged to the centre, only by the attraction of the inner sphere, in the surface of which it is placed: but this force is proportional to the radius of the sphere, or to the distance of the particle from the centre (11.).

13. Let PNQ and ABC (fig. 7.) be two spheres of the same homogeneous matter, which attracts in the inverse proportion of the square of the distance: let the centres of the spheres be at M and D, and take MR equal to the radius of the sphere ABC, and ED equal to the radius of the sphere PNQ; the attractions of the spheres upon particles placed at R and E are to one another as the squares of the radii of the spheres.

In the spheres draw two great circles perpendicular to the diameters PQ and AC, that pass through the points R and E; and let PpQ and AbC be two great circles, making equal indefinitely small angles NMp and BD6 with the great circles PNQ and ABC. Let HK and FG, parallel to PQ and AC, be any two chords of the circles PpQ and AbC, that subtend similar arcs, or arcs containing the same number of degrees; and through HK and FG, let planes perpendicular to the circles PpQ and AbC, be drawn to cut the portions of the spheres contained in the angles NMp and BD6: join RH, RK, MH, MK, DF, DG, EF, EG. Because the arcs subtended by HK and FG, are like parts of their circumferences, it is plain, that the angle RMH = EDF, and RMK = EDG. And because ED = MH = MK, and RM = DF = DG (hyp.), therefore RH = EF, and RK = EG.

Conceive the chords HK and FG, together with the planes passing through them, to change their place a little, so as to describe two slender prisms, or elements of the portions of the spheres contained in the angles NMp and BD6. It is plain that MR and DO, the distances of the chords HG and FG from the centres of their circles, are constantly proportional to MN and DB, the radii of the spheres; wherefore RT and OS, the perpendicular sections of the small prisms, are similar figures, and have to one another the same ratio that MX² has to DO², or MN² to DB². Now, the attraction of the prism HK urging a particle at R to the centre M, is

\[ XT \times \left\{ \frac{1}{RH} - \frac{1}{RK} \right\} \] (10.); and the attraction of the prism FG urging a particle at E to the centre D, is

\[ OS \times \left\{ \frac{1}{EF} - \frac{1}{EG} \right\}. \]

But, in consequence of what was proved,

\[ \frac{1}{RH} - \frac{1}{RK} = \frac{1}{EF} - \frac{1}{EG}; \]

wherefore the attractions of the prisms are to one another as XT to OS, or as MN² to DB².—The same thing may be proved of all the elements of the two portions of the spheres contained in the angles NMp and BD6; wherefore those portions attract particles at R and E with forces proportional to the squares of the radii of the spheres. But because the small angles NMp and BD6 are equal, each of the spheres may be divided into an equal number of such portions; wherefore the attractions of the whole spheres upon particles placed at R and E, are proportional to the squares of the radii of the spheres.

Cor. This proposition is true, when the particles of matter attract one another with forces proportional to any proposed function of the distance.

Let RH = EF = f, and RK = EG = f'; then, adopting the same notation as before (8. Cor.), the attractions of the prisms urging particles placed at R and E to the centres M and D, are respectively XT \( \times \left\{ \Psi(f) - \Psi(f') \right\} \), and OS \( \times \left\{ \Psi(f) - \Psi(f') \right\} \); consequently, those attractions have the same pro- portion that RT has to SO, or MN² to DB². Wherefore the attractions of the whole spheres are in the same proportion.

14. A particle placed anywhere without a sphere of homogeneous matter which attracts in the inverse proportion of the square of the distance, will be urged to the centre of the sphere, with a force that is inversely proportional to the square of the particle's distance from the centre.

Let ABC (Plate XXX. fig. 8.) be the sphere, O its centre, and P a particle without the sphere: conceive a concentric sphere PMN, of the same homogeneous matter with the sphere ABC, to be described with the radius PO. Then, by the last proposition, the attraction of the sphere ABC upon the particle P, is to the attraction of the sphere PMN upon a particle placed at A, as AO² to PO². But the attraction of the sphere PMN upon a particle placed at A, is equal to the attraction of the sphere ABC upon the same particle; for the attraction of the matter between the two spherical surfaces exerts no force upon a particle at A (9.). Therefore, in the proportion set down above, the two middle terms are constantly the same wherever the point P is placed without the sphere ABC. Consequently, the first term of the proportion must follow the inverse ratio of the last term; that is, the attraction of the sphere ABC upon the external particle at P is inversely proportional to PO².

15. The same law of attraction being supposed, a homogeneous sphere will attract a particle placed without it, with the same force as if all the matter of the sphere were collected in the centre.

Let f denote the distance of the particle from the centre; then it follows, from the last proposition, that the attraction of the sphere upon the particle will have, for its measure, \( \frac{A}{f^2} \); A denoting a constant quantity that will be determined by any particular case; that is, by the actual attractive force corresponding to any determinate distance from the centre. Let r denote the radius of the sphere, and M its mass; then no part of the matter of the sphere being nearer the attracted particle than \((f-r)\), and none of it more remote than \((f+r)\), the attraction of the sphere on the particle will be greater than \( \frac{M}{(f-r)^2} \), and less than \( \frac{M}{(f+r)^2} \). Therefore \( \frac{A}{f^2} \) is always contained between those limits, which requires that \( A = M \). For, if A were greater than M, such values of f might be found as would make \( \frac{A}{f^2} \) equal to, or greater than \( \frac{M}{(f-r)^2} \); and, if A were less than M, such values of f might be found as would make \( \frac{A}{f^2} \) equal to, or less than \( \frac{M}{(f+r)^2} \).

Therefore \( A = M \); and the attraction of the sphere is equal to \( \frac{M}{f^2} \), or the same as if all the matter were collected in the centre.

If the radius of the sphere \( = r \), the density of the Attraction matter contained in it \( = d \); then the mass, or \( M = \frac{4\pi r^3d}{3} \) (\( \pi \) being the circumference of the circle whose diameter is unit), and the attraction of the sphere at the distance \( f \) from the centre \( = \frac{4\pi r^3d}{3f^2} \). This is still true at the surface of the sphere when \( f = r \), so that the attraction at the surface \( = \frac{4\pi rd}{3} \); which expression, with the help of what is proved in (12.), enables us to compare the intensities of the attractions of homogeneous spheres, at all distances from the centre, without or within the surfaces.

Cor. 1. A shell of homogeneous matter contained between two concentric spherical surfaces, will attract a particle placed without it, with the same force as if all the matter of the shell were collected in its centre.

For the attractive force of such a shell is equal to the difference of the attractions of two concentric spheres of the same homogeneous matter with the shell.

Cor. 2. A sphere composed of concentric shells, that vary in their densities according to any law, will attract a particle placed without it, with the same force as if all the matter were collected in the centre.

For this having been proved of one shell (Cor. 1.), it must be true of any number of shells.

If \( \varphi(r) \) denote the density at the distance \( r \) from the centre, the quantity of matter in the sphere will be \( = \int_0^r \varphi(r) r^2 dr \); and the attraction on a particle without the sphere at the distance \( f \) from the centre \( = \frac{4\pi f \varphi(r) r^2 dr}{f^2} \).

16. Two spheres, each composed of concentric shells of variable density, attract one another with the same force as if all the matter of each were collected in its centre.

For the attraction of a sphere A upon every particle of another sphere B will remain the same, if we suppose all the matter of A to be collected in its centre (15.). But the attraction of any particles of matter placed in A's centre, upon the sphere B, is equal and opposite to the attraction of B, upon the same matter so placed: and, again, the attraction of B upon all the particles placed in the centre of A, will remain unchanged, if we suppose the matter of B to be collected in its centre. Wherefore A attracts B with the same force as if the matter of each were collected in its centre.

17. Supposing that the particles of matter attract with a force proportional to the distance, a body of any shape will attract a particle of matter placed anywhere with the same force, and in the same direction, as if all the matter of the body were collected in its centre of gravity.

Suppose that the attracted particle is placed at P (fig. 9.), and the centre of gravity of the attracting body at G; join PG, and let any plane pass through that line. Let L be a small part, or element of the body, and Attraction. From L draw LK perpendicular to the plane passing through PG, and KF perpendicular to PG; join PL and PK. Put dm to denote the quantity of matter, or the mass of the element L; then its attractive force, urging the particle in the direction PL, is \( PL \times dm \), which, by the resolution of forces, is equivalent to the two forces, \( PK \times dm \) and \( KL \times dm \); and, again, the single force \( PK \times dm \) is equivalent to the two forces \( FK \times dm \) and \( PF \times dm = PG \times dm + GF \times dm \). Therefore, the attraction of the element L, upon the particle at P, is equivalent to these four separate forces, viz. \( PG \times dm \), \( GF \times dm \), \( FK \times dm \), \( KL \times dm \), which urge the particle P respectively in the directions, PG, GF, FK, KL. But, from the nature of the centre of gravity, the sum of all the forces, \( KL \times dm \), that urge the particle P to one side of the plane passing through PG, is just equal to the sum of the forces that urge it to the other side of the same plane; and the sum of all the forces, \( FK \times dm \), that urge P to one side of the line PG, is just equal to the sum of the forces that urge it to the other side of the same line; and the sum of all the forces, \( GF \times dm \), that urge P towards the point G, is just equal to the sum of the forces that urge it from the same point. Therefore all the preceding forces mutually destroy one another, excepting the forces, \( PG \times dm \), the sum of which, when extended to all the elements of the attracting body, is \( = PG \times \text{mass of the body} \). Therefore the whole attraction upon P is the same as if all the matter of the body were collected in its centre of gravity.

Cor. Supposing that the particles of matter attract with a force proportional to the distance, a homogeneous sphere will attract a particle placed anywhere in the same manner as if all the matter of the sphere were collected in the centre.

For the centre of gravity of a homogeneous sphere, is the same as the centre of its figure. This corollary is likewise true of a sphere composed of concentric shells of variable density; and it is easy to apply the demonstration of (16.) to prove that, in this law of attraction, two spheres, each composed of concentric shells of variable density, will attract one another with the same force as if the matter of each were collected in its centre.

18. To investigate what are the laws of attraction, in regard to the distance, according to which a shell of homogeneous matter, contained between two concentric spherical surfaces, will attract a particle placed without it, in the same manner as if all the matter of the shell were collected in the centre.

It has been proved that this property actually belongs to homogeneous shells in the law of attraction which obtains in nature, and likewise when the particles of matter attract with a force proportional to the distance; but it is interesting to know whether it is confined to these two cases alone, or extends to other laws of attraction. This can only be discovered by a direct analysis.

Let \( r = PC \) (fig. 10.), the distance of the attracted point from the centre of the shell; \( u = CA \) the radius of the inner surface of the shell; \( f = PM \), the distance of P from any point in the surface. Having drawn the diameter AD through P, let AMD and AND be two great circles, making, with one another, an indefinitely small angle MAN = \( dq \); and let two Attraction, small circles BMG, bmg, indefinitely near one another, of which A and D are the poles, meet the former circles in M, N, m, n; and draw MS, NS to the centre of the circle BMG. Put \( \delta \) for the measure of the arc AM; then \( MS = u \sin \delta \); \( MN = dq \cdot u \sin \delta \); \( MM = u^2 dq \cdot d\delta \sin \delta \); and the quadrilateral space \( MNmm = u^2 dq \cdot d\delta \sin \delta \). We may suppose the thickness of the shell indefinitely small; since, if the property belong to an elementary shell indefinitely thin, it will be true of one of a determinate thickness, which can be resolved into such elements. Suppose the thickness of the shell to be \( = du \); then, the quantity of matter in the part standing upon the quadrilateral space \( MNmm = u^2 du \cdot dq \cdot d\delta \sin \delta \). Let \( \varphi(f) \) represent the direct attraction of a particle at M in the direction PM; then its attraction directed to the centre C \( = \varphi(f) \times \frac{PS}{PM} = \frac{r - u \cos \delta}{f} \times \varphi(f) \); and the attraction of the element of the shell in the same direction \( = u^2 du \cdot dq \cdot d\delta \sin \delta \times \frac{r - u \cos \delta}{f} \cdot \varphi(f) \).

This expression is proportional to \( dq \), when \( \delta \) and \( f \) remain constant; and, therefore (denoting by \( \sigma \) the circumference of the circle whose diameter is unit), the attraction of the whole zone contained between the small circles BMG, bmg, will be \( = 2 \pi u^2 du \cdot d\delta \sin \delta \cdot \frac{r - u \cos \delta}{f} \cdot \varphi(f) \); and the attractive force of the whole shell will be

\[ 2 \pi u^2 du \int d\delta \sin \delta \cdot \frac{r - u \cos \delta}{f} \cdot \varphi(f) \]

the fluent to be extended from \( \delta = 0 \) to \( \delta = \pi \).

Again, the quantity of matter in the shell is \( = 4 \pi u^2 du \); and the attraction of this matter placed in the centre, at the distance \( r \) from P, is \( = 4 \pi u^2 du \cdot \varphi(r) \).

If now we equate the attraction of the shell, to the attraction of its matter placed in the centre, and leave out the factors common to both, we shall get

\[ 2 \pi u^2 \int d\delta \sin \delta \cdot \frac{r - u \cos \delta}{f} \cdot \varphi(f) \]

the limits of the integral being the same as before.

But \( f^2 = r^2 - 2ru \cos \delta + u^2 \); then \( d\delta \sin \delta = \frac{fd\delta}{ru} \); also \( r - u \cos \delta = \frac{f^2 + r^2 - u^2}{2r} \); wherefore, by substitution, we get

\[ 4 \pi u^2 \varphi(r) \cdot u = \int (f^2 + r^2 - u^2) \cdot df \cdot \varphi(f) \]

or, which is equivalent, \( 4 \pi u^2 \varphi(r) \cdot u = \int (f^2 + r^2 - u^2) \cdot df \cdot \varphi(f) - 2 \int fdf \cdot \int df \cdot \varphi(f) \),

the limits of this integral being from \( f = r - u \) to \( f = r + u \), which correspond to \( \delta = 0 \) and \( \delta = \pi \).

Now let \( \int df \cdot \varphi(f) = \psi(f) \); and \( \int fdf \cdot \int df \cdot \varphi(f) = \int fdf \cdot \psi(f) = \psi'(f) \); then, by taking the fluents between the proper limits, we get Attraction.

\[ \varphi(r) = 2r \left\{ (r+u)^n - (r-u)^n \right\} - 2 \left\{ r^n + u^n \right\}. \]

If we develope the binomial functions in the last expression, all the even powers of \( u \) will disappear, and the odd powers only will remain; these last terms being all contained in this general formula,

\[ \text{viz. } \frac{4}{1,2,3 \ldots 2n+1} \left\{ r \cdot \frac{d^{2n+1}}{dr^{2n+1}} \varphi(r) \right\}^n. \]

and, observing that \( \frac{d}{dr} \varphi(r) = r \varphi(r) \), the same expression will become

\[ \frac{4}{1,2,3 \ldots 2n+1} \left\{ r \cdot \frac{d^{2n+1}}{dr^{2n+1}} \varphi(r) \right\}^n. \]

which, again, is more simply expressed thus, viz.

\[ \frac{4}{1,2,3 \ldots 2n+1} \left\{ r \cdot \frac{d^{2n+1}}{dr^{2n+1}} \varphi(r) \right\}^n. \]

Wherefore, by substituting the development instead of the functions, and then, by dividing by \( 4r^2 \), we get

\[ \varphi(r) = \frac{d}{dr} \varphi(r) + \frac{1}{1,2,3} \left\{ \frac{1}{r} \cdot \frac{d^2}{dr^2} \varphi(r) \right\}^n. \]

From the nature of the function \( \varphi(r) \), we get \( \varphi(r) = \frac{d}{dr} \varphi(r) \); therefore each of the remaining terms must be separately equal to nothing: Hence

\[ \frac{d}{dr} \left\{ \frac{1}{r} \cdot \frac{d^2}{dr^2} \varphi(r) \right\} = 0; \]

from which we find \( r \varphi(r) = \frac{1}{2} A r^2 + A' r - A'' \),

\( A, A', A'' \) being arbitrary constant quantities; and this value of \( r \varphi(r) \), it is plain, will likewise render all the succeeding terms of the development evanescent. Therefore

\[ \varphi(r) = \frac{d}{dr} \varphi(r) = A r + \frac{A'}{r}. \]

Thus the most general expression of the law of attraction, that possesses the property in question, is a combination of the two laws above mentioned, with each of which it coincides, according as we make the one or other of the constant quantities equal to nothing. We have therefore a direct proof, that the law of nature is the only one which will make the attraction decrease as the distance increases, and in which a spherical shell, or a sphere, will attract in the same manner as if all the matter were collected in the centre.

Laplace has arrived at the same conclusion by a different process. (Mech. Celeste, Liv. 2d. Chap. 2. No. 12. Rem. Part.)

Attraction of Spheroids of Revolution.

19. Let \( APBQ \) (Plate XXXI, fig. 11.) and \( CMDN \) be two concentric ellipses, similar to one another, and similarly situated, of which \( AB \) and \( CD \) are either the greater, or less, axes; and let \( PCQ \) be perpendicular to \( AB \). Conceive the ellipse to revolve about \( PQ \) so as to describe an indefinitely small angle; then, supposing the law of attraction to be inversely proportional to the square of the distance, the thin solid of homogeneous matter described by the ellipse \( APBQ \) will attract a particle placed at \( P \), in a direction perpendicular to any plane passing through \( PQ \), with the same force that the thin solid of the same matter described by the ellipse \( CMDN \), will attract a particle placed at \( C \) perpendicularly to the same plane.

From \( C \) draw \( CM, CN \), making equal angles with \( CD \), and \( PR, PT \) respectively parallel to \( CM, CN \); and let \( Cm, Ca, Pr, Pt \) be drawn in the same manner, and indefinitely near the former lines. While the ellipses revolve about \( PQ \), the small sectors will describe pyramids that have their vertices at \( C \) and \( P \).

It is manifest that the pyramids so described are similar; for their angles at \( C \) and \( P \) in the planes of the ellipses are equal; and their other angles described by revolving about \( PQ \) are likewise equal, because the sectors are equally inclined to that axis. Wherefore, the direct attractions of all the small pyramids upon the particles \( P \) and \( C \), are proportional to the lengths \( PR, PT, CM, CN \); and consequently the forces that urge the particles \( P \) and \( C \) in a direction at right angles to any plane passing through \( PQ \), are proportional to the perpendiculars let fall upon that plane from \( R, T, M, N \). But, because \( PR, PT, CM, CN \) are equally inclined to \( PQ \), they will make equal angles with any plane passing through \( PQ \); wherefore the perpendiculars drawn to the plane from \( R, T, M, N \), will be respectively proportional to \( CM, CN, PR, PT \). But \( CM + CN = PR + PT \); wherefore, the sum of the perpendiculars drawn to the plane from \( M \) and \( N \), will be equal to the sum of the perpendiculars drawn to it from \( R \) and \( T \). Consequently the force of the pyramids \( PR \) and \( PT \), which urges the particle \( P \) at right angles to the plane, is equal to the force of the pyramids \( CM \) and \( CN \), which urges the particle \( C \) in a parallel direction. The same thing is true of all the small pyramids that make up the thin solids described by the ellipses \( APBQ \) and \( CMDN \); and it is therefore true of the whole solids.

It is to be observed, that when the pyramids \( PR \) and \( PT \) fall on opposite sides of \( PQ \), it is the difference of their attractions which is equal to the sum of the attractions of \( CM \) and \( CN \); and it is the difference of the perpendiculars let fall from \( T \) and \( R \) on opposite sides of the plane, which is equal to the sum of the perpendiculars let fall from \( M \) and \( N \). 20. Let \( APBQ \) be a spheroid of revolution, \( PQ \) the axis of revolution, and \( ACB \) a plane through the centre perpendicular to \( PQ \) (Plate XXXI, fig. 12.). If \( D \) be a particle in the surface of the spheroid, and \( DL \) perpendicular to the plane \( ACB \); then the attraction of the spheroid on a particle placed at the pole \( P \), will be to the force with which a particle placed at \( C \), is attracted in the direction \( DL \), as \( PC \) is to \( DL \).

Through \( D \) draw a plane parallel to the plane \( ACB \), and let the plane so drawn cut the axis \( PQ \) in \( F \): draw the straight line \( DFE \) to terminate in the spheroid, and describe another spheroid through \( F \), having the same centre with the spheroid \( APBQ \), and similar to it, and similarly situated. Conceive an indefinitely great number of planes, making indefinitely small angles with one another, to be drawn through \( DE \), so as to divide the two spheroids into an indefinitely great number of thin solids, or slices; then the sections which every one of the planes makes with the spheroids will be similar ellipses, having the same centre (\( s_1 \)); and it is manifest that a straight line drawn through \( F \) at right angles to \( DE \), in any one of the planes, will pass through the centre of the two ellipses contained in it, and will coincide with an axis of each. Therefore, the force with which every one of the slices, or elements, of the spheroid \( APBQ \) attracts a particle placed at \( D \) in the direction \( DL \), is equal to the force with which the corresponding slice, or element, of the spheroid \( GFHK \), attracts a particle placed at \( F \) in the direction \( FC \) (19.). Therefore, the whole attraction of the spheroid \( APBQ \) upon a particle at \( D \), in the direction \( DL \), is equal to the whole attraction of the spheroid \( GFHK \), upon a particle at \( F \). But the attractions of the spheroids \( APBQ \) and \( GFHK \), upon particles placed at \( P \) and \( F \), are to one another as \( PC \) to \( FC \) (8.). Therefore, the attraction of the spheroid \( APBQ \) upon a particle at \( P \), is to the force with which the same spheroid attracts a particle at \( D \), in the direction \( DL \), as \( PC \) is to \( FC \) or \( DL \).

21. Let \( APBQ \) be a spheroid of revolution, and \( PQ \) the axis of revolution, as before. If \( D \) be a particle in the surface, \( ADPB \) (fig. 13.) a section through \( D \), and the axis \( PQ \), and \( DL \) perpendicular to \( PQ \); the attraction of the spheroid upon a particle at \( A \), will be to the force with which a particle at \( D \) is attracted, in the direction \( DL \), as \( AC \) is to \( DL \).

Through \( D \) draw a plane perpendicular to \( AB \), which cuts the section \( ADB \) in the straight line \( DFE \); and let a spheroid \( FGHK \) be described through \( F \), having the same centre with the spheroid \( APBQ \), and similar to it, and similarly situated. Then, conceiving the two spheroids to be divided into an indefinitely great number of thin slices by planes passing through \( DE \), the force with which every slice, or element, of the spheroid \( APBQ \) attracts a particle at \( D \) in the direction \( DL \), will be equal to the force with which the corresponding slice, or element, of the spheroid \( FGHK \) attracts a particle at \( F \) (19.). But the attractions of the spheroids \( APBQ \) and \( FGHK \) upon particles placed at \( A \) and \( F \), are to one another as \( AC \) to \( CF \) (8.). Therefore, the attraction of the spheroid \( APBQ \) upon a particle at \( A \), is to the force with which the same spheroid attracts a particle at \( D \), in the direction \( DL \), as \( AC \) to \( FL \) or \( DL \).

The two last propositions will enable us to find both the direction and the intensity of the attraction of a homogeneous spheroid of revolution upon a particle placed anywhere on the surface, when we have ascertained the attractive forces at the poles, and at the circumference of the circular section made by a plane through the centre perpendicular to the axis. For the whole attraction at any point, is the compound force arising from the attractions perpendicular to the axis, and parallel to it. The next object of our research is, therefore, to determine the two forces above-mentioned, viz. the attraction at the poles, and at the circular section, equally distant from both poles.

22. Let \( ADB \) be an indefinitely slender pyramid, of which the base \( BD \) is perpendicular to the edge \( AD \) (fig. 14.): let \( B = \text{base } BD \), and \( f = \text{length } AD \); then \( \frac{B}{f} \) is the attraction of the whole matter of the pyramid upon a particle placed at the vertex \( A \).

Let \( AM = x \); then the section \( MN \) parallel to the base \( BD = \frac{Bx^2}{f^2} \); and, \( MP = \text{element of the prism} = \frac{Bx^2dx}{f^2} \); and the attraction of the element upon a particle placed at \( A = \frac{MP}{AM^2} = \frac{Bdx}{f^2} \); the fluent of which is \( \frac{Bx}{f^2} = \text{attraction of the pyramid } AM \) upon a particle at \( A \). And, when \( x = f \), this becomes \( \frac{B}{f} = \text{attraction of the pyramid } AD \) upon a particle placed at \( A \).

23. To investigate the attraction of a homogeneous spheroid of revolution, upon a particle placed at the pole.

Let \( P \) (fig. 15.) be the pole, \( PCQ \) the axis of revolution, and \( APBQ \) a section of the spheroid by a plane passing through \( PQ \), and any point \( M \), in the surface; draw \( PM, PM' \) indefinitely near \( PM \), and \( MM' \) perpendicular to \( PM \). Conceive the plane \( PMQ \) to revolve about \( PQ \), so as to describe the indefinitely small angle \( BCO \); then the small triangle \( MMP' \) will describe a slender pyramid, having its vertex at \( P \), and of which the base is a rectangle, contained by \( MM' \) and \( RT \); for the point \( M \) moving parallel to \( R \), it will describe a line equal and parallel to that described by \( R \), namely, to \( RT \).

Let \( PM = f \); and the angle \( KPM \), which \( PM \) makes with a perpendicular to the axis, \( = \phi \); and the indefinitely small angle \( BCO = d\phi \). Then \( MM' = f d\phi \); \( RT = CR \times d\phi = f \cos \phi \times d\phi \); and \( B \), the base of the slender pyramid described by the triangle \( MMP' \), \( = d\phi \times d\phi \cos \phi \times f^2 \); wherefore, the direct attraction of the pyramid on a particle at \( P = \frac{B}{f} \) (22)= Attraction. \( d\varphi \cdot d\theta \cos \varphi \cdot f \); and the elementary attraction of the spheroid in the direction PC = direct attraction of the pyramid \( \times \frac{PS}{PM} = d\varphi \cdot d\theta \cos \varphi \cdot f \).

Again, let MR = x, CR = y, PC = k, AC = k'; then \( y = f \cos \varphi; x = k - f \sin \varphi \); if we substitute these values in the equation of the solid (1), we get

\[ \frac{(k - f \sin \varphi)^2}{k^2} + \frac{f^2 \cos^2 \varphi}{k^2} = 1; \quad \text{whence} \]

\[ f = \frac{2k^2}{k^2} \cdot \frac{\sin \varphi}{\cos^2 \varphi + \frac{k^2}{k^2} \sin^2 \varphi}. \]

By substituting the value of \( f \) just found in the preceding expression of the elementary attraction of spheroid, it will become

\[ \frac{2k^2}{k^2} \cdot \frac{d\varphi \cdot d\theta \cdot \sin \varphi}{\cos^2 \varphi + \frac{k^2}{k^2} \sin^2 \varphi}; \]

which must be integrated from \( \varphi = 0 \) to \( \varphi = 2\pi \);

and from \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \); \( \sigma \) denoting always the half-circumference when radius is unit.

In an oblate spheroid \( k \) is less than \( k' \); put \( k'^2 - k^2 = k^2 e^2 \), and \( z = \sin \varphi \); then the element of the attractive force will become, by substitution,

\[ \frac{2k^2}{k^2} \cdot \frac{dz}{1 + e^2 z^2} = \frac{2k^2}{k^2} \cdot \left( \frac{edz}{1 + e^2 z^2} \right); \]

and by integrating from \( z = 0 \) to \( z = 1 \), we get,

\[ \frac{2k^2}{k^2} \cdot \left\{ e - \text{arc. tan. } e \right\}; \]

for the force with which the matter between the planes PBQ and POQ urges the particle P to the centre. Therefore the whole attractive force of the spheroid upon a particle at P is

\[ \frac{4\pi k^2}{k^2 e^2} \left\{ e - \text{arc. tan. } e \right\}; \]

And, because \( \frac{4\pi k^2}{3} = \text{mass of the spheroid} = M \),

we get the measure of the attraction of the oblate spheroid upon a particle placed at the pole, equal to

\[ k \cdot \frac{3M}{k^2 e^2} \left\{ e - \text{arc. tan. } e \right\}. \]

In an oblong spheroid, \( k \) is greater than \( k' \); put \( k'^2 - k^2 = k^2 e^2 \); then the element of the attractive force will become, by substitution,

\[ \frac{2k^2}{k^2} \cdot \frac{dz}{1 - e^2 z^2} = \frac{2k^2}{k^2} \cdot \left( \frac{edz}{1 - e^2 z^2} - edz \right); \]

whence, by proceeding as before, we get the measure of the attractive force of the oblong spheroid on a particle placed at the pole, equal to

\[ k \cdot \frac{3M}{k^2 e^2} \left\{ \frac{1}{2} \text{hyp.log. } \frac{1 + e}{1 - e} \right\}. \]

Cor. In an oblate spheroid differing little from a sphere, \( e^2 \) will be a very small fraction, of which we may reject the higher powers. When this is done, Attraction the preceding expression of the polar attraction, viz.

\[ \frac{4\pi k^2}{k^2 e^2} \left( e - \text{arc. tan. } e \right), \]

will be \( 4\pi k \cdot (1 + e^2)(\frac{1}{2} - \frac{1}{2} e^2) = \frac{4\pi k}{3} \left( 1 + \frac{2}{5} e^2 \right) \).

And, if \( k' = k + r = k \sqrt{1 + e^2} \), be the radius of the equator, then \( \frac{r}{k} = e^2 \); so that the attraction at the pole will be

\[ \frac{4\pi k}{3} \left( 1 + \frac{4}{5} \cdot \frac{e^2}{k} \right). \]

24. To investigate the attraction of a homogeneous spheroid of revolution, on a particle placed in the circumference of the circular section, made by a plane through the centre, at right angles, to the axis of revolution.

Let P (Plate XXXI. fig. 16.) be the pole, PC the axis of revolution. A, a point in the circular section AOB, made by a plane through the centre perpendicular to PC. Let M be any point in the surface of the spheroid; AMO a section through A and M by a plane perpendicular to AOB; AM a line in that plane indefinitely near AM, and MM perpendicular to AM; MR perpendicular to AO, and RS to AB. Conceive the plane AMO to revolve about A, so as to describe an indefinitely small angle OAM; then the triangle AMR will describe a slender pyramid, having its vertex at A, and of which the base is equal to a rectangle contained by MM and RT; for the point M moving parallel to the point R, it will describe a line equal to that described by R, namely to RT.

Let AM = \( f \); the angle MAR = \( \varphi \); and the angle OAQ = \( d\varphi \); then MM = \( f d\varphi \); and TR = AR \( \times \) \( d\varphi \) \( = f \cos \varphi \cdot d\varphi \). Wherefore, B = base of the pyramid described by MM = \( d\varphi \cdot d\theta \cos \varphi \cdot f^2 \); and the direct attraction of the pyramid in the direction AM = \( \frac{B}{f} \)

\( = d\varphi \cdot d\theta \cos \varphi \cdot f \). Therefore, the elementary attraction of the spheroid, in the direction AC = direct attraction of the pyramid \( \times \frac{AR}{AM} \times \frac{AS}{AR} = d\varphi \cos \varphi \cdot d\theta \cos \varphi \cdot f \cdot f \).

Again, let MR = x, RS = y, CS = z, CP = k and AC = \( k' \); then (1.)

\[ \frac{x^2}{k^2} + \frac{y^2}{k^2} + \frac{z^2}{k^2} = 1. \]

But \( x = f \sin \varphi; y = AR \cdot \sin \varphi = f \cos \varphi \cdot \sin \varphi \); and \( z = k' - f \cos \varphi \cdot \cos \varphi \); therefore, by substitution, we get

\[ \frac{f^2 \sin^2 \varphi}{k^2} + \frac{f^2 \cos^2 \varphi \sin^2 \varphi}{k^2} + \frac{(k' - f \cos \varphi \cdot \cos \varphi)^2}{k^2} = 1. \]

From this equation we get

\[ f = 2k \cdot \frac{\cos \varphi \cdot \cos \varphi}{\cos^2 \varphi + \frac{k^2}{k^2} \sin^2 \varphi}. \]

Let this value of \( f \) be substituted in the expression of the elementary attraction of the spheroid before found, and it will become which expression, when integrated from $\phi = 0$ to $\phi = \pi$, and from $z = 0$ to $z = 1$, will give the attraction of half the spheroid; and the double of it, viz.

$$4k' \cdot d\phi \cos^2 \phi \cdot \frac{dz}{\cos^2 \phi + k'^2 \sin^2 \phi},$$

being integrated between the same limits, will give the whole attraction of the spheroid.

In the oblate spheroid, $k$ is less than $k'$; Let $k'^2 = k^2 - k'^2 e^2$, and $z = \sin \theta$; and, by substitution, the element of the attractive force will become

$$4k' \cdot d\phi \cos^2 \phi \cdot \frac{dz(1 - z^2)}{1 + e^2 z^2} =$$

$$4k' \cdot d\phi \cos^2 \phi \cdot \frac{1 + e^2}{e^2} \left\{ \frac{dz}{1 + e^2 z^2} - \frac{dz}{1 + e^2} \right\}.$$

And, by integrating from $z = 0$ to $z = 1$, we get

$$4k' \cdot d\phi \cos^2 \phi \cdot \frac{1 + e^2}{e^2} \left\{ \text{arc.tan.e} - \frac{e}{1 + e^2} \right\},$$

for the force with which the matter between the sections that contain the angle OAQ, attracts the particle A to the centre. But

$$\int d\phi \cos^2 \phi = \frac{\pi}{2} (1 + \cos^2 \phi) = \frac{\pi}{2} + \frac{1}{4} \sin 2\phi,$$

the value of which, between the limits $\phi = 0$, and $\phi = \pi$, is $\frac{\pi}{2}$;

wherefore, the attraction of the spheroid on a particle at A, is equal to

$$2\pi \cdot k' \cdot \frac{1 + e^2}{e^2} \left\{ \text{arc.tan.e} - \frac{e}{1 + e^2} \right\}.$$

Because

$$\frac{4\pi k'^2 k}{3} = \frac{4\pi k^2 (1 + e^2)}{3} = M,$$

we get

$$2\pi \cdot (1 + e^2) = \frac{3M}{2k^2}:$$

wherefore, the measure of the attractive force of the oblate spheroid on a particle placed anywhere in the circumference of the circular section made by a plane through the centre at right angles to the axis, is equal to

$$k' \cdot \frac{3M}{2k^2 e^2} \left\{ \text{arc.tan.e} - \frac{e}{1 + e^2} \right\}.$$

In the oblong spheroid, $k$ is greater than $k'$; put $k'^2 = k^2 - k'^2 e^2$: then the element of the attractive force will become, by substitution,

$$4k' \cdot d\phi \cos^2 \phi \cdot \frac{dz(1 - z^2)}{1 - e^2 z^2} = 4k' \cdot d\phi \cos^2 \phi \cdot \frac{1 - e^2}{e^2} \left\{ \frac{dz}{1 - e^2 z^2} - \frac{dz}{1 - e^2} \right\},$$

whence, by proceeding as before, we get the measure of the attractive force of the oblong spheroid upon a particle placed anywhere in the circumference of the circular section, made by a plane through the centre at right angles to the axis, equal to

$$k' \cdot \frac{3M}{2k^2 e^2} \left\{ \text{arc.tan.e} - \frac{e}{1 + e^2} \right\}.$$

Cor. In an oblate spheroid, differing little from a sphere, the higher powers of $e^2$ may be neglected. The expression of the attractive force at the equator, viz.

$$2\pi \cdot k' \cdot \frac{1 + e^2}{e^2} \left\{ \text{arc.tan.e} - \frac{e}{1 + e^2} \right\},$$

will then become

$$2\pi \cdot k' \cdot (1 + e^2) \left( \frac{2}{3} - \frac{4}{5} e^2 \right) = \frac{4\pi k'}{3} \cdot (1 - \frac{1}{5} e^2).$$

And if $k'$, the radius of the equator, $= k + r$; then

$$\frac{2\pi}{k} = e^2 (23, Cor.)$$

and the attraction at the equator will be equal to

$$\frac{4\pi k}{3} \cdot (1 + \frac{r}{k}) \cdot \left( 1 - \frac{2}{5} \frac{r}{k} \right) = \frac{4\pi k}{3} \cdot \left( 1 + \frac{3}{5} \frac{r}{k} \right).$$

25. An oblate spheroid of revolution being given, it is required to find the measures of the attractive forces that urge a particle placed anywhere in the surface, in a direction perpendicular to the axis, and in a direction parallel to it.

Let $k$ and $k'$ be the semiaxes of the ellipse by the revolution of which the spheroid is described, $k$ being the axis about which it revolves; and let $b$ be the perpendicular distance of the particle from the axis, and $a$ its distance from the plane, drawn through the centre at right angles to the axis: Then, from which was proved in (20,) and (21,), the attractions sought will be found by multiplying the attractions at the pole, and at the circular section equally distant from both poles, by $\frac{a}{k}$ and $\frac{b}{k'}$.

Thus we get the attraction in the direction of $a$, equal to

$$a \times \frac{3M}{k e^2} \left\{ e - \text{arc.tan.e} \right\};$$

and the attraction in the direction of $b$, equal to

$$b \times \frac{3M}{2k^2 e^2} \left\{ \text{arc.tan.e} - \frac{e}{1 + e^2} \right\}.$$

The same formulæ likewise serve for finding the attractions upon a particle placed anywhere within the spheroid. For the attraction upon a particle within the spheroid is equal to the attraction of a similar concentric spheroid, which contains the particle in its surface (9.) and it is evident, that the coefficients, which multiply $a$ and $b$ in the above expressions, depend only upon the proportion of $k$ and $k'$; and they are therefore the same for all similar spheroids.

If we denote by $A$ and $B$ the coefficients of $a$ and $b$ in the expressions of the attractive force found above, the whole attraction of the spheroid, which is compounded of the forces $a \cdot A$ and $b \cdot B$, will be $\sqrt{a^2 A^2 + b^2 B^2}$. And if $\psi$ denote the angle which the direction of this force makes with $a$, or with the axis of the spheroid; then tan. $\psi = \frac{b.B}{a.A}$. Attraction.

Cor. In the very same manner we may determine the attractions of an oblong spheroid of revolution, upon a point in the surface, or within the solid.

26. If \( k, k', k'' \), the semiaxes of a homogeneous ellipsoid, be related to those of another ellipsoid of the same matter, \( h, h', h'' \), so that \( k^2 - k'^2 = h^2 - h'^2 \) and \( k''^2 - h''^2 = h^2 - h'^2 \), the attractions perpendicular to the planes of the principal sections, which the first ellipsoid (Plate XXXI., fig. 17.) exerts upon a point determined by the coordinates \( k \sin m, h' \cos m \sin n, h'' \cos m \cos n \), respectively parallel to \( k, k', k'' \), will be to the attractions which the second ellipsoid exerts upon a point determined by the coordinates \( k \sin m, h' \cos m \sin n, h'' \cos m \cos n \), respectively parallel to \( h, h', h'' \), in the direct proportion of the areas of the principal sections to which the attractions are perpendicular.

This proposition is an extension to all elliptical spheroids of what was proved of the sphere in (13.). It is here enunciated of the ellipsoid, because the demonstration is not more difficult for that solid than for spheroids of revolution.

Let \( ABDM \) be an ellipsoid, the semiaxes of which are \( BC = k, EC = k', AC = k'' \); and \( acdm \) another ellipsoid, of which the semiaxes are \( bc = h, ec = h', ac = h'' \); those quantities being so related, that \( k^2 - k'^2 = h^2 - h'^2 \), and \( k''^2 - h''^2 = h^2 - h'^2 \). Also, let \( G \) be a point about the ellipsoid \( ABDM \), so determined that \( GH \), parallel to \( BC, = h \sin m; HK \), parallel to \( CE, = h' \cos m \sin n; \) and \( CK = h'' \cos m \cos n \); and let \( g \) be a point about the ellipsoid \( acdm \), so determined that \( gh \), parallel to \( bc, = k \sin m; hk, \) parallel to \( ce, = k' \cos m \sin n; \) and \( ck = k'' \cos m \cos n \). Then the force with which the ellipsoid \( ABDM \) attracts a particle placed at \( G \) in the direction \( GH \), will be to the force with which the ellipsoid \( acdm \) attracts a particle placed at \( g \) in the direction \( gh \), as the area of the section \( AEDM \) to the area of the section \( acdm \), or as \( k' k'' \) to \( h' h'' \).

Let \( RP = k \sin \phi; PO = k' \cos \phi \sin \varphi; \) and \( CO = k'' \cos \phi \cos \varphi; \) which suppositions are allowable, because they satisfy the equation of the ellipsoid (1.), whatever be the angles \( \phi \) and \( \varphi \). Draw \( CPM \) through the centre, and \( CN \) indefinitely near it; then \( CP = \cos \phi \sqrt{k^2 \sin^2 \varphi + k'^2 \cos^2 \varphi} \); and when \( \cos \phi = 1, CM = \sqrt{k^2 \sin^2 \varphi + k'^2 \cos^2 \varphi} \); wherefore \( \frac{CP}{CM} = \cos \phi \). Let the angle \( DCM = \psi \); then \( \tan \psi = \frac{PO}{CO} = \frac{k'}{k''} \tan \varphi \); and, by taking the fluxions,

\[ \frac{d^2 \psi}{\cos^2 \psi} = \frac{k'}{k''} \cdot \frac{d\varphi}{\cos^2 \varphi}; \quad \text{but} \quad \frac{1}{\cos^2 \psi} = 1 + \tan^2 \psi = \frac{k'^2 \sin^2 \varphi + k''^2 \cos^2 \varphi}{k'^2 \cos^2 \varphi} = \frac{CM^2}{k'^2 \cos^2 \varphi}; \]

wherefore \( d\psi \cdot CM^2 = \text{twice the sector } MCN = k' k'' \cdot d\varphi \).

And, in like manner, in the other ellipsoid, if \( rp = h \sin \phi; po = h' \cos \phi \sin \varphi; \) and \( co = h'' \cos \phi \cos \varphi; \) then \( \frac{CP}{CM} = \cos \phi \), and twice the sector \( mcn = h' h'' \cdot d\varphi \).

It is plain, from what has been shown, that, when Attraction, \( \phi \) varies, and \( \varphi \) remains constant in the expressions of the coordinates, the points \( P \) and \( p \) will move along \( CM \) and \( cm \), so that, in every position, \( \frac{PC}{MC} = \frac{po}{mc} \).

Let \( Q \) and \( q \) be indefinitely near \( P \) and \( p \); and through \( P \) and \( Q \) draw lines parallel to \( MN \); and through \( p \) and \( q \) draw lines parallel to \( mn \). Let \( S \) denote the quadrilateral contained between the parallels drawn through \( P \) and \( Q \); and \( S' \) that contained between the lines drawn through \( p \) and \( q \): Then

\[ S = \frac{MCN}{MC^2} = \frac{QC^2 - PC^2}{MC^2}; \quad \text{and} \quad S' = \frac{qC^2 - pc^2}{mc^2}; \]

wherefore, since \( \frac{PC}{MC} = \frac{po}{mc} \) and \( \frac{QC}{MC} = \frac{qc}{mc} \), it is manifest that

\[ S = \frac{MCN}{mc^2} = \frac{k' k''}{h' h''}. \]

Upon the quadrilaterals \( S \) and \( S' \) let upright prisms \( RS \) and \( rs \) be erected, and be prolonged to meet the surfaces of the spheroids; join \( GR, GS, gr, gs \). Then,

\[ GR^2 = (h \sin m - k \sin \phi)^2 + (h' \cos m \sin n - k' \cos \phi \sin \varphi)^2; \]

\[ + (h'' \cos m \cos n - k'' \cos \phi \cos \varphi)^2; \]

\[ g r^2 = (k \sin m - h \sin \phi)^2 + (k' \cos m \sin n - h' \cos \phi \sin \varphi)^2; \]

\[ + (k'' \cos m \cos n - h'' \cos \phi \cos \varphi)^2. \]

And, by expanding these expressions, we get

\[ GR^2 = \left\{ (h^2 + (k'^2 - h'^2) \cos^2 m \sin^2 n + (h''^2 - h'^2) \cos^2 m \cos^2 n) + (k^2 + (k'^2 - k^2) \cos^2 \phi \sin^2 \varphi + (k''^2 - k'^2) \cos^2 \phi \cos^2 \varphi) \right\}; \]

\[ gr^2 = \left\{ (k^2 + (k'^2 - k^2) \cos^2 m \sin^2 n + (k''^2 - k'^2) \cos^2 m \cos^2 n) + (h^2 + (h'^2 - h^2) \cos^2 \phi \sin^2 \varphi + (h''^2 - h'^2) \cos^2 \phi \cos^2 \varphi) \right\}; \]

These expressions are equal, because \( k'^2 - k^2 = h'^2 - h^2 \), and \( k''^2 - k'^2 = h''^2 - h'^2 \); wherefore \( RG = gr = f \). And, in like manner, it is shown, that \( GS = gs = f' \).

Now, the attraction of the prism \( RS \) urging a particle at \( G \) in the direction \( GH \), is equal to

\[ S \times \left\{ \frac{1}{f} - \frac{1}{f'} \right\} \quad (10); \]

and the attraction of the prism \( rs \) urging a particle at \( g \) in the direction \( gh \), is \( S' \times \left\{ \frac{1}{f} - \frac{1}{f'} \right\} \); wherefore these attractions Attraction.

The same thing may be proved of all the elementary prisms that make up the two portions of the spheroids contained between the planes BGM, BCN, and bcn, ben; therefore, those portions attract particles at G and g, with forces proportional to $k'k''$ and $k''k'$. But the two spheroids may be divided into an equal number of such portions; therefore the spheroids attract particles placed at G and g, in the directions GH and gh, with forces proportional to $k'k''$ and $k''k'$, or to the sections AMDE and amde.

Cor. 1. This proposition is true when the law of attraction is expressed by any function of the distance. The demonstration is the same as in the corollary of (13).

Cor. 2. If the two ellipsoids be so placed, that their centres, and the planes of their principal sections, shall coincide, the surface of the one will be entirely within the other. Also the point which one ellipsoid attracts, will be in the surface of the other, as is plain from the expressions of the coordinate. And hence, the attraction of one ellipsoid upon a point without the surface, is made to depend upon the attraction of another ellipsoid upon a point within the surface.

Cor. 3. When the ellipsoids become spheroids of revolution, the two principal sections through the axis of revolution become equal, and will be represented by any two sections whatever passing through the axis at right angles to one another. But, in this case, the attractions of the spheroids on the points may be reduced to two, one acting perpendicular to the axis, and one parallel to it: And it is plain, that these attractions will be to one another as the areas of the sections, perpendicular to their directions.

27. To find the attraction of an oblate spheroid upon a particle placed without the surface.

Let $k'$ be the radius of the equator, and $k$ the axis of revolution: and let $a$ be the perpendicular distance of the point without the spheroid from the plane of the equator, and $b$ its distance from the axis. In the first place, it is necessary to determine the semiaxis of another oblate spheroid that shall contain the given point in its surface, and such, that it shall have the same centre, and its equator in the same plane, as the given spheroid; and likewise, the difference of the squares of its semiaxes equal to the difference of the squares of the semiaxes of the given spheroid. Let $k'$ denote the radius of the equator, and $h$ the semiaxis of the required spheroid: then, because the attracted point is to be in the surface of the solid, we have

$$\frac{a^2}{k'^2} + \frac{b^2}{k^2} = 1$$

and, because

$$k'^2 - k^2 = k'^2 - k^2 = v^2$$

we get

$$\frac{a^2}{k'^2} + \frac{b^2}{k^2} = 1$$

Whence,

$$2k^2 = a^2 + b^2 - v^2 + \sqrt{(a^2 + b^2 - v^2)^2 + 4a^2v^2}$$

and when $h$ is determined, then $k' = \sqrt{k'^2 + v^2}$.

In consequence of the equation $\frac{a^2}{k'^2} + \frac{b^2}{k^2} = 1$, we may suppose, $a = k \sin m$, and $b = k' \cos m$; let $a' = k \sin m$, and $b' = k' \cos m$; or $a' = \frac{k}{h} a$, and $b' = \frac{k'}{k} \times b$: then the point determined by the coordinates $a'$ and $b'$ will be in the surface of the given spheroid, and, consequently, it will be within the surface of the other spheroid. Let $M'$ denote the mass of the spheroid of which the axis is $h$; and let

$$e'^2 = \frac{k'^2 - k^2}{k^2} = \frac{k'^2 - k^2}{k^2}$$

then (25.) the attractions of this spheroid upon the point within its surface, determined by the coordinates $a$ and $b$, are these, viz.

That perpendicular to the equator, equal to

$$a' \times \frac{3M'}{k^2 e'^2} \left\{ e' - \text{arc. tan. } e' \right\}$$

and that perpendicular to the axis, equal to

$$b' \times \frac{3M'}{2k^2 e'^2} \left\{ \text{arc. tan. } e' - \frac{e'}{1 + e'^2} \right\}$$

But (26. Cor. 3.) the attractions of the given spheroid, whose semiaxis are $k$ and $k'$ upon the point without its surface determined by the coordinates $a$ and $b$, will be found by multiplying the preceding expressions respectively by $\frac{k'^2}{k^2}$ and $\frac{k^2}{k'^2}$. Let $M$ be the mass of the given spheroid; then

$$M = \frac{k'^2 k}{M'}$$

consequently

$$\frac{k'^2}{k^2} = \frac{M}{M'} \cdot \frac{h}{k} = \frac{M}{M'} \cdot \frac{a}{a'}$$

and $\frac{k^2}{k'^2} = \frac{M}{M'} \cdot \frac{h}{k} = \frac{M}{M'} \cdot \frac{b}{b'}$: Wherefore, the attractions of the given oblate spheroid upon a point, without the surface determined by the coordinates $a$ and $b$, are as follows, viz.

The attraction perpendicular to the equator, equal to

$$a \times \frac{3M}{k^2 e'^2} \left\{ e' - \text{arc. tan. } e' \right\}$$

and that perpendicular to the axis, equal to

$$b \times \frac{3M}{2k^2 e'^2} \left\{ \text{arc. tan. } e' - \frac{e'}{1 + e'^2} \right\}$$

Cor. In the very same manner we may determine the attractions of an oblong spheroid of revolution upon a point without the surface.

Attractions of Ellipsoids.

28. Let AMBN be one of the principal sections of an ellipsoid, C the centre, AB and MN the axes, D a point in the periphery of the section, and DO perpendicular to MN (Plate XXXI., fig. 18.); the attraction of the ellipsoid upon a particle placed at the pole A, is to the force with which a particle placed at D is attracted in the direction DO, as AC to DO. Attraction. Draw DFG perpendicular to AB, and through F describe an ellipsoid similar to the given ellipsoid, and similarly situated, and having the same centre. Conceive an indefinitely great number of planes making indefinitely small angles with one another, to be drawn through DG, so as to divide the two ellipsoids into an indefinitely great number of thin solids or slices; Then, the sections of the ellipsoids made by every one of the planes will be similar and concentric ellipses, each of them having an axis perpendicular to DG (5.). Wherefore the attractions of the elements of the ellipsoid FHKL upon a particle at F, are respectively equal to the attractions of the elements of the ellipsoid AMBN, upon a particle at D in the direction DO (19.). Therefore, the whole attraction of the ellipsoid FHKL upon a particle at F, is equal to the attraction of the ellipsoid AMBN upon a particle at D, in the direction DO. But the attractions of the ellipsoids AMBN, and FHKL upon particles at A and F, are to one another as AC to CF (8.). Therefore, the attraction of the ellipsoid AMBN upon a particle at the pole A, is to the force with which it attracts a particle at D in the direction DO, as AC to DO.

29. The attractions of ellipsoids upon particles placed in the surface, urging them in directions perpendicular to any of the principal sections, are proportional to the distances of the particles from that section.

Let AMBN be one of the principal sections of an ellipsoid, C the centre, AB and MN the axes of the section, and P a point in the surface of the solid; the attraction of the ellipsoid upon a particle at the pole A (Plate XXXI. fig. 19.), is to the force with which a particle at P is attracted in a direction parallel to AB, as the semiaxis AC is to the distance of P from the principal section perpendicular to AC.

Draw PDQ perpendicular to the section AMBN, and let it meet the surface again in Q; through D describe an ellipsoid similar to AMBN, similarly situated, and having the same centre; and through P draw a section SPRQ perpendicular to AB. As before, divide the solids into an indefinitely great number of thin slices by planes drawn through PQ; the sections made by every one of those planes will be similar, and concentric ellipses having an axis of each perpendicular to PQ (5.). Wherefore, the attractions of the elements of the ellipsoid AMBN, upon a particle at P, in a direction perpendicular to the plane PRQS, are respectively equal to the attractions of the elements of the ellipsoid FHKL, upon a particle at D, in a direction perpendicular to the same plane (19.). Therefore, the attraction of the ellipsoid AMBN, upon a particle at P, in a direction parallel to the axis AB, is equal to the attraction of the ellipsoid FHKL, upon a particle at D in the same direction. But the ellipsoids AMBN and FHKL being similar, their attractions upon particles at A and F, are to one another as AC to CF (8.) ; and the attraction of the ellipsoid FHKL, upon a particle at the pole F, is to its attraction upon a particle at D, in a direction parallel to AC, as FC to CN (28.). Therefore (ex aequali), the attraction of the ellipsoid AMBN, upon a particle at the pole A, is to the force with which it attracts a particle at P, in the direction AC, as AC to CN.

This proposition will enable us to find the attractions of an ellipsoid on all points on the surface, or Attraction within the solid, when the attractions at the poles are determined.

30. To investigate the differential expressions of the attractions at the poles of an ellipsoid.

Let APD be an ellipsoid; C the centre; AC, CE, and PC, the semiaxes; and PMB a section made by a plane through PC and any point, M in the surface; draw PM (fig. 20.) PM indefinitely near PM, and MM perpendicular to PM; also MR perpendicular to the plane ADB, MS perpendicular to PC, and RH perpendicular to AD. Conceive the plane PCB to revolve about PC, so as to describe an indefinitely small angle BCO; and let PM = f; the angle KPM, which PM makes with a perpendicular to the axis, = θ; and the angle DGB = ϕ: then by proceeding as in (23.), it will be found that the attraction of the small pyramid described by the triangle MPm, urging a particle at P to the centre of the ellipsoid, is = dp dθ cos θ sin θ f.

Again, let MR = x, HR = y, CII = z; also let PC = k, AC = k', CE = k"; then x = k - f sin θ; y = f cos θ sin ϕ; z = f cos θ cos ϕ; and if we substitute these values in the equation of the ellipsoid (1.), we shall get

\[ \frac{(k-f \sin \theta)^2}{k^2} + \frac{f^2 \cos^2 \theta \sin^2 \phi}{k'^2} + \frac{f^2 \cos^2 \theta \cos^2 \phi}{k''^2} = 1 \]

whence

\[ f = \frac{k}{k'^2} \cdot \frac{2 \sin \theta}{\sin^2 \theta + \cos^2 \theta \sin^2 \phi + \cos^2 \theta \cos^2 \phi} \]

This is the value of f at the pole of k; and, by a like procedure, its values at the poles of k' and k" may be found, viz.

\[ f = \frac{k'}{k'^2} \cdot \frac{2 \sin \theta}{\sin^2 \theta + \cos^2 \theta \sin^2 \phi + \cos^2 \theta \cos^2 \phi} \]

Suppose that k is the least of the semiaxes; and let \(k^2 = \frac{k^2}{m}\) and \(k'^2 = \frac{k^2}{n}\); then the values of f at the poles of k, k', k", will be, respectively,

\[ f = \frac{2 k \sin \theta}{\sin^2 \theta + m \cos^2 \theta \sin^2 \phi + n \cos^2 \theta \cos^2 \phi} \]

Now, let A, A', A", denote the attractions of the spheroid upon particles placed at the poles of k, k', k"; then, by substituting the values of f just found in the foregoing differential expression, we get

\[ A = k \times \int \int \frac{2 dp d\theta \cos \theta \sin \theta}{\sin^2 \theta + m \cos^2 \theta \sin^2 \phi + n \cos^2 \theta \cos^2 \phi} \] Attraction.

\[ A' = k' \times \int \int \frac{m \times 2 dp \cdot d \cos \theta \sin^2 \phi}{\sin \theta + \cos \theta \sin^2 \phi + n \cos \theta \cos^2 \phi} \]

\[ A'' = k'' \times \int \int \frac{n \times 2 dp \cdot d \cos \theta \sin^2 \phi}{\sin \theta + \cos \theta \sin^2 \phi + m \cos \theta \cos^2 \phi} \]

the limits of the integrals being from \( \theta = 0 \) and \( \phi = 0 \) to \( \theta = \frac{\pi}{2} \) and \( \phi = 2\pi \).

31. To reduce the expressions of the polar attractions to the most simple integrals.

Let us consider the general expression

\[ \int \int \frac{d \phi \cdot d \theta \cos \theta \sin^2 \phi}{a \sin^2 \theta + b \cos^2 \theta \sin^2 \phi + c \cos^2 \theta \cos^2 \phi} \]

which includes all the formulas found in (30.) Let \( p = a \sin^2 \theta + b \cos^2 \theta \); and \( q = a \sin^2 \phi + c \cos^2 \phi \); then the above expression will become

\[ \int \int \frac{d \phi \cdot d \theta \cos \theta \sin^2 \phi}{p \sin^2 \phi + q \cos^2 \phi}. \]

Suppose \( \sqrt{\frac{p}{q}} \cdot \frac{\sin \phi}{\cos \phi} = \frac{\sin u}{\cos u} \); then the preceding expression will become, by substitution,

\[ \int \int \frac{du \cdot d \theta \cos \theta \sin^2 \phi}{\sqrt{p \cdot q}}; \]

the limits of \( u \) being from 0 to \( 2\pi \); therefore, by integrating with regard to \( u \), and restoring the values of \( p \) and \( q \), the integral becomes

\[ 2 \pi \int \frac{d \theta \cdot \cos \theta \sin^2 \phi}{(a \sin^2 \theta + b \cos^2 \theta)(a \sin^2 \phi + c \cos^2 \phi)} \]

and, by putting \( x = \sin \theta \), the integral, which is to be taken from \( \theta = 0 \) to \( \theta = \frac{\pi}{2} \), or from \( x = 0 \) to \( x = 1 \), will become

\[ 2 \pi \int \frac{x^2 dx}{\sqrt{b + (a - b)x^2} \cdot \sqrt{c + (a - c)x^2}}. \]

If now we take \( a, b, c \) so as to make the assumed expression coincide with the quantities \( A, A', A'' \), respectively, we shall get

\[ A = 4\pi k \int \frac{x^2 dx}{\sqrt{m + (1-m)x^2} \cdot \sqrt{n + (1-n)x^2}} \]

\[ A' = 4\pi k' \int \frac{m \cdot x^2 dx}{\sqrt{1 + (m-1)x^2} \cdot \sqrt{n + (m-n)x^2}} \]

\[ A'' = 4\pi k'' \int \frac{n \cdot x^2 dx}{\sqrt{1 + (n-1)x^2} \cdot \sqrt{m + (n-m)x^2}}. \]

These expressions have the inconvenience of containing different factors in the denominators; but they may be reduced to others having the same factors, by putting \( x = \frac{r}{\sqrt{m + (1-m)x^2}} \) in the second, and \( x = \frac{r}{\sqrt{n + (1-n)x^2}} \) in the third; we thus get

\[ A = 4\pi k \int \frac{x^2 dx}{\sqrt{m + (1-m)x^2} \cdot \sqrt{n + (1-n)x^2}} \]

\[ A' = 4\pi k' \int \frac{m \cdot x^2 dx}{\sqrt{1 + (m-1)x^2} \cdot \sqrt{n + (m-n)x^2}} \]

\[ A'' = 4\pi k'' \int \frac{n \cdot x^2 dx}{\sqrt{1 + (n-1)x^2} \cdot \sqrt{m + (n-m)x^2}}. \]

Now let

\[ \frac{1-m}{m} = \frac{k^2 - k'^2}{k^2} = \lambda^2; \text{ and } \frac{1-n}{n} = \frac{k'^2 - k''^2}{k'^2} = \lambda'^2. \]

Also let the mass of the ellipsoid be \( M = \frac{4\pi k' k''}{3 \sqrt{mn}} \); then \( \frac{3M}{k^3} = \frac{4\pi}{\sqrt{mn}} \): wherefore, by substitution, we get

\[ A = k \cdot \frac{3M}{k^3} \int \frac{x^2 dx}{\sqrt{(1 + \lambda^2 x^2) \cdot (1 + \lambda'^2 x^2)}} \]

\[ A' = k' \cdot \frac{3M}{k'^3} \int \frac{x^2 dx}{\sqrt{1 + \lambda^2 x^2} \cdot \sqrt{1 + \lambda'^2 x^2}} \]

\[ A'' = k'' \cdot \frac{3M}{k''^3} \int \frac{x^2 dx}{\sqrt{1 + \lambda^2 x^2} \cdot \sqrt{1 + \lambda'^2 x^2}}. \]

The integrations extending from \( x = 0 \) to \( x = 1 \).

These integrals cannot be expressed in finite terms. When \( \lambda \) and \( \lambda' \), or the eccentricities of the ellipsoid are small, the values of the integrals may easily be found to a sufficient degree of exactness, by series.—They may likewise be all expressed by means of this fluent, viz.

\[ F = \int \frac{dx}{\sqrt{(1 + \lambda^2 x^2) \cdot (1 + \lambda'^2 x^2)}} \text{ (from } x = 0 \text{ to } x = 1 \text{) and its partial fluxions. Thus we have, in general, } \int \frac{x^2 dx}{\sqrt{(1 + \lambda^2 x^2) \cdot (1 + \lambda'^2 x^2)}} = \]

\[ \frac{x^3}{\sqrt{(1 + \lambda^2 x^2) \cdot (1 + \lambda'^2 x^2)}} + \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right). \]

Wherefore, making \( x = 1 \), we get

\[ A = k \cdot \frac{3M}{k^3} \cdot \frac{1}{\sqrt{(1 + \lambda^2) \cdot (1 + \lambda'^2)}} + \]

\[ \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right). \]

\[ A' = k' \cdot \frac{3M}{k'^3} \cdot \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right). \]

\[ A'' = k'' \cdot \frac{3M}{k''^3} \cdot \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right). \]

32. To find the forces with which a homogeneous ellipsoid attracts a particle placed in the surface, or within the solid, in directions perpendicular to the principal sections.

Let \( k, k', k'' \), denote the semiaxes of an ellipsoid, Attraction, and \(a, b, c\) (respectively parallel to \(k, k', k''\)), the perpendicular distances of a particle placed in the surface, or within the solid, from the principal sections; then, from what is proved in (29.), the attractions we are seeking will be found by multiplying the polar attractions by \(\frac{a}{k}, \frac{b}{k'}, \frac{c}{k''}\). Therefore the forces that urge the particle in the directions of \(a, b,\) and \(c\), are respectively,

\[a \times \frac{3M}{k^3} \left\{ \frac{1}{\sqrt{(1 + \lambda^2)(1 + \lambda'^2)}} + \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right) \right\};\]

\[b \times \frac{3M}{k'^3} \times \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right);\]

\[c \times \frac{3M}{k''^3} \times \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right);\]

Which formulas serve both for points in the surface, and within the solid, for the reason already explained in (25.).

33. To find the attractions of an ellipsoid upon a particle placed without the surface.

Let \(k, k', k''\) be the semiaxis of the ellipsoid, and \(a, b, c\) (respectively parallel to \(k, k', k''\)), the coordinates of a particle without the surface. Let \(h, h', h''\), so related to \(k, k', k''\), that \(h^2 = k^2 - k'^2\) and \(h'^2 = k'^2 - k''^2\), denote the semiaxis of another ellipsoid, which contains the attracted point in its surface, and has its principal sections in the same planes as the given ellipsoid; then, because the attracted point is in the surface, we have

\[\frac{a^2}{h^2} + \frac{b^2}{h'^2} + \frac{c^2}{h''^2} = 1;\]

And, because \(h'^2 = k'^2 - k''^2 = s^2\), and \(h''^2 = k''^2 - k'^2 = t^2\), we get

\[\frac{a^2}{h^2} + \frac{b^2}{s^2} + \frac{c^2}{t^2} = 1.\]

This equation now contains only one unknown quantity; and it is plain, that one value of \(h\), and only one, can be found from it. For, when \(h = 0\), the function on the left hand side is infinitely great; And while \(h\) increases from 0 ad infinitum, the same function decreases continually from being infinitely great, to be infinitely little. When \(h\) is found, then \(h' = \sqrt{h^2 + s^2}\), and \(h'' = \sqrt{h^2 + t^2}\). Because \(a, b, c\), are the coordinates of a point in the surface of the ellipsoid, we may suppose \(a = h \sin m, b = h' \cos m \sin n, c = h'' \cos m \cos n\); let \(a' = k \sin m, b' = k' \cos m \sin n, c' = k'' \cos m \cos n\); or \(a' = \frac{k}{h} \times a\),

\[b' = \frac{k'}{h'} \times b, c' = \frac{k''}{h''} \times c;\] then \(a', b', c'\) will be the coordinates of a point in the surface of the given ellipsoid, and consequently, it will be within the other solid. Let \(M'\) denote the mass of the ellipsoid of which \(h, h', h''\) are the semiaxes; also let \(\lambda^2 = \text{Attraction.}\)

\[\frac{h'^2}{h^2} = \frac{k'^2 - k''^2}{k^2}; \quad \text{and} \quad \lambda'^2 = \frac{h''^2}{h'^2} = \frac{k''^2 - k'^2}{k'^2};\]

then, \(F\) denoting the same fluent as before, the attractions of this ellipsoid upon the point within it, determined by the coordinates \(a', b', c'\), in the directions of those coordinates, are (32.) respectively equal to

\[a' \times \frac{3M'}{h^3} \left\{ \frac{1}{\sqrt{(1 + \lambda^2)(1 + \lambda'^2)}} + \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right) \right\};\]

\[b' \times \frac{3M'}{h'^3} \times \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right);\]

\[c' \times \frac{3M'}{h''^3} \times \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right).\]

Now, the attractions of the given ellipsoid upon the point without the surface, determined by the coordinates \(a, b, c\), will be found (26.) by multiplying the preceding expressions respectively by \(\frac{k}{h}, \frac{k'}{h'}, \frac{k''}{h''}, \frac{k}{h'}\). Let \(M\) be the mass of the given ellipsoid; then

\[\frac{M}{M'} = \frac{k}{h} \frac{k'}{h'} \frac{k''}{h''}; \quad \text{consequently} \quad \frac{k'}{h'} = \frac{M}{M'} \frac{h}{k} = \frac{M}{M'} \frac{a}{a'};\]

\[\frac{b}{b'} = \frac{M}{M'} \frac{b}{b'}; \quad \text{and} \quad \frac{k''}{h''} = \frac{M}{M'} \frac{c}{c'};\]

wherefore, the attractions of the given ellipsoid upon the point without the surface, determined by \(a, b, c\), in the directions of those coordinates, are respectively equal to

\[a \times \frac{3M}{k^3} \times \left\{ \frac{1}{\sqrt{(1 + \lambda^2)(1 + \lambda'^2)}} + \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right) + \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right) \right\};\]

\[b \times \frac{3M}{k'^3} \times \frac{1}{\lambda} \left( \frac{dF}{d\lambda} \right);\]

\[c \times \frac{3M}{k''^3} \times \frac{1}{\lambda'} \left( \frac{dF}{d\lambda'} \right).\]

The preceding propositions contain a complete theory of homogeneous elliptical spheroids. They enable us to compute the attractive force with which a solid of this kind urges a particle placed anywhere in the surface, or within the solid, or without it. It remains, indeed, to find the exact value of the function \(F\) in its general form, to which we can do no more than approximate by series; but this is an analytical difficulty, which it is impossible to overcome; because the nature of this function is such, that it cannot be expressed in finite terms by the received notation of analysis. In the preceding investigations, we have followed the method of Maclaurin for points situated in the surface of a spheroid, or within the solid. This method has always been justly admired; but neither its inventor, nor, as far as we know, any other Geometer, has applied it, excepting to spheroids of revolution; and it is here, for the first time, extended to ellipsoids. In regard to points without the surface, we have employed the method first given by Mr Ivory, in the Philosophical Transactions for 1809. The combination of these two methods has enabled us to derive the attractions of an ellipsoid on a point placed anywhere, from the attractions at the poles. Thus, this extremely complicated problem has, by geometrical reasoning of no great difficulty, been reduced to the investigation of the polar attractions, which are the only cases that require a direct computation.

**Attraction of Mountains.** See Mountains, Attraction of, in the Encyclopedia, and in this Supplement.

**Atwood (George),** an Author celebrated for the accuracy of his mathematical and mechanical investigations, and considered as particularly happy in the clearness of his explanations, and the elegance of his experimental illustrations, was born in the early part of the year 1746. He was educated at Westminster school, where he was admitted in 1759. Six years afterwards he was elected off to Trinity College, Cambridge. He took his degree of Bachelor of Arts in 1769, with the rank of third wrangler, Dr Parkinson, of Christ's College, being senior of the year. This distinction was amply sufficient to give him a claim to further advancement in his own College, on the list of which he stood foremost of his contemporaries; and, in due time, he obtained a fellowship, and was afterwards one of the tutors of the College. He became Master of Arts in 1772; and, in 1776, was elected a Fellow of the Royal Society of London.

The higher branches of the Mathematics had, at this period, been making some important advances at Cambridge, under the auspices of Dr Waring, and many of the younger members of the University became diligent labourers in this extensive field. Mr Atwood chose, for his peculiar department, the illustration of mechanical and experimental philosophy, by elementary investigations and ocular demonstrations of their fundamental truths. He delivered, for several successive years, a course of lectures in the Observatory of Trinity College, which were very generally attended, and greatly admired. In the year 1784, some circumstances occurred which made it desirable for him to discontinue his residence at Cambridge; and soon afterwards Mr Pitt, who had become acquainted with his merits by attending his lectures, bestowed on him a patent office, which required but little of his attendance, in order to have a claim on the employment of his mathematical abilities in a variety of financial calculations, to which he continued to devote a considerable portion of his time and attention throughout the remainder of his life.*

The following, we believe, is a correct list of Mr Atwood's publications:

1. *A Description of Experiments to illustrate a Course of Lectures.* 8vo. About 1775, or 1776. 2. This work was reprinted with additions, under the title of *An Analysis of a Course of Lectures on the Principles of Natural Philosophy.* 8vo. Cambr. 1784. 3. *A General Theory for the Measurement of the Angle subtended by two objects, of which one is observed by Rays after two Reflections from plane Surfaces, and the other by Rays coming directly to the Spectator's Eye.* Phil. Trans. 1781, p. 395. 4. *A Treatise on the Rectilinear Motion and Rotation of Bodies, with a Description of Original Experiments relative to the Subject.* 8vo. Cambr. 1784. 5. *Investigations founded on the Theory of Motion, for determining the Times of Vibration of Watch Balances.* Phil. Trans. 1794, p. 119. 6. *The Construction and Analysis of Geometrical Propositions, determining the positions assumed by homogeneous bodies, which float freely, and at rest, on a fluid's surface; also Determining the Stability of Ships, and of other Floating Bodies.* Phil. Trans. 1796, p. 46. 7. *A Dissquisition on the Stability of Ships.* Phil. Trans. 1798, p. 201. 8. *A Review of the Statutes and Ordinances of A-size, which have been established in England from the 4th year of King John, 1202, to the 27th of his present Majesty.* 4to. Lond. 1801. 9. *A Dissertation on the Construction and Properties of Arches.* 4to. Lond. 1801. 10. *A Supplement to a Tract entitled a Treatise on the Construction and Properties of Arches, published

---

*See Literary Memoirs of Living Authors, 2 vol. 8vo, Lond. 1798; Morning Herald, 17th July 1807; Nichols's Literary Anecdotes of the Eighteenth Century, Vol. VIII. 8vo, Lond. 1814.* in the year 1801; and containing Propositions for Determining the weights of the several sections which constitute an arch, inferred from the angles. Also containing a Demonstration of the angles of the several sections, when they are inferred from the weights thereof. To which is added, a Description of original experiments to verify and illustrate the principles in this treatise. With occasional remarks on the construction of an iron bridge of one arch, proposed to be erected over the river Thames at London. Part II. By the author of the first part. 4to, Lond. 1804. Dated 24th Nov. 1803.

11. A Treatise on Optics is mentioned by Nichols as having been partly printed by Bowyer in 1776, but never completed.

It may be very truly asserted, that several of these works of Mr Atwood have materially contributed to the progress of science, by multiplying the modes of illustration, which experimental exhibitions afford for the assistance of the instructor; at the same time, they can scarcely be said to have extended very considerably the bounds of human knowledge, or to demonstrate that their author was possessed of any extraordinary talent or energy of mind in overcoming great difficulties, or in inventing new methods of reasoning. The Analysis of a Course of Lectures has been little read: and it bears evident marks of having been composed before Mr Atwood had acquired a habit of accurate reasoning on Physical subjects. In the first page, for example, the forces of cohesion and gravitation are completely confounded; and in the third we find the idea of perfect spheres touching each other in a greater or less number of points, notwithstanding the appearance of precision which the author attempts to maintain in his language.

The object of the paper on Reflection is, to illustrate and improve the construction of Hadley's quadrant; and Mr Atwood proposes, for some particular purposes in practical Astronomy, two new arrangements of the speculums, by which the rays are caused to move in different planes, and which he considers as affording greater accuracy for the measurement of small angles than the common form of the instrument, although not of general utility, nor very easily adjusted for observation.

The treatise on Rectilinear motion and rotation exhibits a good compendium of the elementary doctrines of mathematical Mechanics; but it shows a great deficiency in the knowledge of the higher refinements which had been introduced into that science by Daniel Bernoulli, and Euler, and Lagrange. The properties of simply accelerated and retarded motion are first discussed, and the phenomena of penetration experimentally examined; the laws of varying forces are then investigated, and the properties of the pendulum demonstrated; the vibrations of an elastic chord are calculated, "considering the whole mass to be concentrated in the middle point," as an approximation; and then, instead of imitating and simplifying the elegant but complicated demonstrations of the continental mathematicians, the author most erroneously repeats, in the words of Dr Smith, the exploded doctrine, that "the string, during any instant of its vibration, will coincide with the harmonic curve." The subject of a resistance, varying as the square of the velocity, is next examined; and some useful experiments on the descent of bodies in water are stated in confirmation of the theory. On this occasion, the author observes, with regard to the formation of the different strata of the earth, that bodies disposed to break into large masses, though specifically lighter, may easily have descended more rapidly through a fluid, than denser but more brittle bodies, so that the natural order of densities may thus have become inverted. He next examines the theory of rotation, and relates some very interesting experiments on rectilinear and rotatory motions; and he shows that Emerson and Desaguliers were totally mistaken in asserting "that the momentum produced is always equal to the momentum which produces it." The last section of the work, which is devoted to the subject of free rotation, is the most elaborate of the whole; but it exhibits no material extension of the earlier investigations of the Bernoullis and Professor Vince; nor does it contain the important proposition of Segner, relating to the existence of three axes of permanent rotation, at right angles to each other, in every body, however irregular.

Notwithstanding these partial objections, the work may still, in many respects, be considered as classical. The paper on Watch-balances is principally intended to show the advantages which may be obtained, in Mr Mudge's construction, from the effect of subsidiary springs in rendering the vibrations isochronous, their actions being limited to certain portions of the arc of motion. If the author has here again omitted to follow the Continental Mathematicians in some of their refinements of calculation, it must be confessed that his view of the subject has, in this instance, not only the advantage of simplicity, but also that of a nearer approach to the true practical state of the question, than is to be found in the more complicated determinations which had been the result of the labours of some of his predecessors.

But, whatever may be the merits of these investigations, they appear to be far exceeded in importance by the papers on Ships, the first of which obtained for its author the honour of a Copleian Medal. Its principal object is to show how much the stability of a ship will commonly vary, when her situation, with respect to the horizon, is materially altered; and how far the assumptions of theoretical writers, respecting many others of the forces concerned in Naval Architecture, will generally differ from the true state of these forces when they actually occur in Seamanship. In the second part of the investigation, some errors of Bouguer and of Clairbois are pointed out, and the theoretical principles of stability are exemplified by a detailed calculation, adapted to the form and dimensions of a particular vessel, built for the service of the East India Company.

The latter years of our author's life do not appear to have been productive of any material advantage to Science. His application to his accustomed pursuits was unremitting; but his health was gradually declining. He had no amusement, except such as was afforded by the continued exercise of his mind, with a change of the object only; the laborious game of chess occupying, under the name of recreation, the hours which he could spare from more productive exertions. He became paralytic some time before his death; and though he partially recovered his health, he did not live to complete his 62d year.

His researches concerning the history of the Assize of Bread must have required the employment of considerable diligence, and some judgment, in the discovery and selection of materials; although certainly the subject was not chosen for the purpose of affording a display of talent. His opinion respecting the operation of the assize, as favourable to the community, may by some be thought to be justified by the want of success which has hitherto attended the experiment on its suspension; but the advocates of that measure would certainly not admit the trial of a year to be sufficient for appreciating its utility.

The title-pages of the works on Arches explain the occasion on which they were brought forwards, and at the same time exhibit a specimen of the want of order and precision which seems to have begun to prevail in the author's faculties: and the works themselves betray a neglect of the fundamental principles of Mechanics, which is inconceivable in a person who had once reasoned with considerable accuracy on mathematical subjects. An anonymous Critic, who is supposed to have been the late Professor Robison (British Critic, Vol. XXI. Jan. 1804), very decidedly, and, at the same time, very respectfully asserted Mr Atwood's error in maintaining, that there was no manner of necessity for the condition, that the general curve of equilibrium of an arch should pass through some part of every one of the joints by which it is divided; and in fact we may very easily be convinced of the truth of this principle, if we reflect that the curve of equilibrium is the true representative of the direction of all the forces acting upon each of the blocks; and that if the whole pressure be anywhere directed to a point situated beyond the limit of the joint, there can be nothing whatever to prevent the rotation of the block on the end of the joint as a centre, until some new position of the block shall have altered the direction of the forces, or until the whole fabric be destroyed. The Critic has also very truly remarked, that the effects of friction have never been sufficiently considered in such arrangements: but a later Author has removed a considerable part of this difficulty in an anonymous publication, by showing, that no other condition is required for determining these effects, than that every joint should be perpendicular to the direction of the curve of equilibrium, either accurately, or within the limit of a certain angle, which is constant for every substance of the same kind, and which he has termed the angle of repose.

In the appointment which enabled Mr Atwood to devote a considerable part of his life to scientific researches, he appears to have had no successor. It was held, and perhaps wisely, that such sinecures have regularly become, in process of time, mere instruments of party interest, instead of being bestowed as encouragements to merit; and it seems to be the invariable maxim of the British Government, that talents deserve no protection, unless they are immediately employed in the service of the Church or of the State; that ornamental accomplishments repay their possessor by the splendour which they confer on his personal existence, but that, in a commercial country, the actual utility of mental as well as of corporal powers, must be measured by their effects; and that these effects must be of a negotiable kind, in order to have a claim to reward. Other Nations, and other Sovereigns, have often thought and acted differently; and they have, perhaps, obtained a forced growth of Science or of Literature, which has contributed, in some degree, to the embellishment of their age; but where the native forest tree acquires so often a form at once beautiful and magnificent, though exposed to all the storms of the seasons, there is little reason to lament the want of the shelter of the Plantation, or of the artificial warmth of the Conservatory. ADDENDUM.

ANNUITIES. As an addition to the article ANNUITIES, we beg to insert here an expeditious method of calculating the values of annuities on single or joint lives, from any tables or bills of mortality, with sufficient accuracy for all practical purposes.

We must begin by determining the mean complement of life, according to the average number of deaths during a certain period, which must vary according to the nature of the proposed calculation; being shorter as the rate of interest is higher; and as the number of lives concerned is greater; but not requiring to be very accurately defined. If the rate of interest be \( r \), we must find the time in which the number of deaths is expressed by the fraction

\[ \frac{3}{r+3} \]

of the whole number of survivors at the given age, for a single life; for two lives, the fraction must be

\[ \frac{3}{r+5} \]

and for three,

\[ \frac{3}{r+7} \]

and, in each of these cases, the time determined from the age of the oldest life must be employed for finding the complements of both the others.

Having thus calculated the complements for each of the ages, we may, in most instances, save ourselves the trouble of further computation, by employing tables of the value of annuities on one and two lives, according to Demoivre's hypothesis. For this purpose, we have only to subtract the complement from 86, and we obtain an equivalent life on this hypothesis. If we take, for example, the age of 20, the number of survivors in the Northampton tables is 5132; and, for a single life, at 3 and at 6 per cent, we must find the time at which they are reduced

\[ \frac{3}{6} \quad \text{and} \quad \frac{3}{9} \]

respectively; that is, to about 2566 and 3421: now at 54 and 43, the numbers are 2530 and 3404; and

\[ \frac{5132 \times 34}{5132 - 2530} = 67.07, \quad \text{and} \quad \frac{5132 \times 23}{5132 - 3404} = 68.3 \]

whence the equivalent ages in Demoivre's tables are 18.93 and 17.7, giving 18.62 and 12.43, for the value of the annuity; while Dr Price's table, deduced from the actual decrements at all ages, gives 18.64 and 12.40.

The utility of this mode of calculation will be still further illustrated by a comparison of the very different values of lives, as indicated by different tables. Taking, for example, the age of 30, and the interest at 5 per cent, we may find the value of the annuity, by this approximation, in different situations, for which correct tables have been published by Dr Price, and may thence infer how much nearer it approaches to the truth than the generality of the results approach to each other:

| Location | Value | |----------|-------| | London | 11.22 | | Northampton | 13.09 | | Sweden, males | 14.04 | | Deparcieux | 14.28 | | Sweden, females | 14.58 |

According to the bills of mortality of London for 1815, out of 9472 survivors at 30, 5573 lived to 50, and this is near enough to \( \frac{3}{8} \) for our purpose; hence the complement is 48.58, and the value of an annuity at 5 per cent. 12-16 years' purchase. Where the age is much greater, the approximation is somewhat less accurate, though not often materially erroneous; thus, at 70, the values, according to the Northampton tables, at 3 and 6 per cent, are 6.23 and 5.35, instead of 6.73 and 5.72 respectively.

In the values of joint lives, there is more difference, according to the different tables employed, than in those of single lives: thus, at 30, the value of an annuity, at 4 per cent. on a single life, differs at Northampton, and in Sweden, in the proportion of 14.78 to 16.00, or of 12 to 13; but, for two joint lives at 30, in that of 11.31 to 12.96, or of 7 to 8; and for three lives, the disproportion would be still greater.

In the absence of Demoivre's tables, or for cases to which they do not extend, it becomes necessary to calculate the value of the annuity for each particular instance. Calling then the complement, as already determined, \( a \), the number of survivors after \( x \) years will be represented by \( a-x \), and the present value of any sum to be paid to each of them by

\[ av^x - xv^x, \quad v \text{ being the present value of a unit payable at the end of a year: and if we suppose such payments to be made continually, their whole present value may be found by multiplying this expression by the fluxion of } x, \text{ and finding the fluent, which will be } -pv^x (a-x-p), \quad p \text{ being } = \frac{1}{Hv}, \text{ or the reciprocal of the hyperbolical logarithm of the amount of a unit after a year. When } x \text{ vanishes, this fluent becomes } -p(a-p), \text{ and when } x = a, p^2v^a; \text{ the difference, divided by } a, \text{ gives the present value of the annuity, } p - \frac{p^2}{a} + \frac{p^2-a}{a}; \text{ from which, when the annuity is supposed to become due and to be paid periodically, we must subtract in all cases half a payment; that is, } \frac{1}{2} \text{ for yearly payments, and } \frac{1}{4} \text{ for quarterly; and if, at the same time, we choose to assume that money is capable of being improved by laying out the interest more frequently than once... a year at the given rate, we must alter the value of \( r \) accordingly.

For two joint lives, the complement of the elder, determined from the fraction \( \frac{3}{r+5} \) being \( a \), and that of the younger, deduced from the deaths in an equal number of years, \( b \), we have for the binary combinations of the survivors, after \( x \) years, \((a-x)(b-x)\) and the fluent will be \(-pv^x(ab-(a+b)x+p)+x^2+2px+2p^2\), which, corrected and divided by \( ab \), gives the value of the annuity \( p - \frac{p^2}{ab}(a+b-2p) - \frac{p^2x}{ab}(a-b+2p) \); and this, with the deduction of half a payment, agrees with the tables calculated on Demoivre's hypothesis, taking the same complements of life.

But for three lives we have no such tables, and this method of calculation becomes therefore of still greater importance. Employing here the fraction \( \frac{3}{r+7} \) for the oldest life, we must determine the complement \( a \) for this life, and those of the two younger, \( b \) and \( c \), from an equal period. The combinations will then be \((a-x)(b-x)(c-x)=abc-(ab+ac+bc)x+(a+b+c)x^2-x^3 \), which we may call \( d-ex+fz^2-x^3 \); hence the fluent is found \(-pv^x(d-e(x+p)+f(x^2+2px+2p^2)-(x^3+3px^2+6p^2x+6p^3)) \).

This, when \( z \) vanishes, becomes \(-p(d-ep+fP^2-6p^3) \), and calling this \(-pg \), the corrected fluent will give the value of the annuity \( \frac{pg}{d} - \frac{pg^2}{d}(g-en+f^2a+2fpa-a^2-3pa^2-6p^2a) \). Thus, if the ages are 10, 20, and 30, and the rate of interest 4 per cent., we find, in the Northampton tables, the survivors at 30 4385, \( \frac{9}{11} \) of which are 3199; and at 40, the survivors are 3170; whence \( a = 57.7 \), and \( b \) and \( c \) found also from periods of 16 years after the respective ages, are 68.5 and 91.7. Calculating with these numbers, we find the value of the annuity 10.954—5 = 10.454. Dr Price's short table gives it 10.438; and Simpson's approximation from the tables of two joint lives 10.563, which is less accurate in this instance, even supposing such tables to have been previously calculated.

It would, indeed, be easy to form, by this mode of computation, a table of the corrections required at different ages for Simpson's approximation, since these corrections must be very nearly the same, whether Demoivre's hypothesis, or the actual decrements of lives be employed, both for the two joint lives, and for the correct determination of the three. But the value thus found would still be less accurate, with respect to any other place, or perhaps even any other time, than the immediate result of the mode of calculation here explained.

It may, perhaps, save some trouble to subjoin a table of the values of \( p \) and their logarithms.

| \( p \) | log. \( p \) | |-------|------------| | 3 | 33.831 | | 4 | 25.497 | | 5 | 20.497 | | 6 | 17.162 |

(NT.) # Table of Articles and Treatises Contained in This Volume

| Abbreviation | Description | |--------------|-------------| | ABBALLATIF, or ABBOLLATIPH. | | | ABERDEENSHIRE. | | | ABERRATION (Spherical). See CATOPTRICS and DIOPTRICS. | | | ABORTION. | | | ABULFEDA, or ABULFEZA. | | | ABSENTEE. | | | ABULFAZEL. | | | ABU-TEMAN. | | | ABBYSSINIA. | | | ACADEMIES. | Military and Naval. | | ACEPHALA. See MOLLUSCA.* | | | ACHROMATIC GLASSES. | | | ACKERMANN (JOHN CHRISTIAN GOTTLIEB). | | | ACOSTA (JOSEPH D'). | | | ACOUSTICS. | | | ADAM (ALEXANDER). | | | ADANSON (MICHAEL). | | | ADELUNG (JOHN CHRISTOPHER). | | | ADEN. | | | ADHESION. | | | ADMIRAL. | Lord High. | | ADMIRALTY, High Court of. | | | ÆPINUS (FRANCIS ULRICH THEODORE). | | | AEROLITE. See METEOROLITE. | | | AERONAUTICS. | | | AFRICA. | | | AFRICAN COMPANY. | Institution. | | AGNESI (MARIA GAETANA). | |

*This reference was omitted in its proper place in the Volume.

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AGRICULTURE. AGUESSEAU (HENRY FRANCIS D'). AIR. See CLIMATE and METEOROLOGY. ALBANIA. ALCARAZAS. See COOLING. ALCOOMETER. ALEUTIAN ISLANDS.