It was remarked in the preceding article, that the decisive experiment by which Pascal established the reality of atmospheric pressure, had likewise suggested to this ingenious philosopher the method of determining the elevations of distant points on the surface of the globe. But the first attempts were very rude, proceeding on the inaccurate supposition that the lower mass of air is a fluid of uniform density. Different authors estimated variously from eighty to ninety feet as the altitude, which corresponds to a variation of the tenth part of an inch in the mercurial column. The Torricellian tube or cane, as it was then called, was, on its first introduction to England, carried accordingly to the tops of mountains, or conveyed to the bottom of pits and mines, or even let down to great depths in the sea.

Among those experimentalists who laboured most assiduously in the study and application of the barometer in this part of the island, we may mention George Sinclair. This ingenious person had been professor of philosophy in the university of Glasgow, but seems to have conscientiously resigned his office soon after the Restoration, rather than comply with that hated episcopacy which the minions of Charles II. had forced upon the people of Scotland. He then retired to the village of Trantent, not far from Edinburgh, and was employed as a practical engineer, in tracing the levels of coal-pits, in directing the machinery employed in the mines at Leadhills, and afterwards in the great undertaking of conducting water from the heights of the Pentlands to supply the northern metropolis. Though not a profound mathematician, he was skilled in mechanics and hydrostatics, and possessed no small share of invention. Sinclair is said to have been the first who applied to the mercurial tube the name of baroscope, or indicator of weight; the more definite appellation of barometer, or measurer of weight, not having been appropriated till many years afterwards. During his excursions in 1688 and 1670, he employed that instrument to measure the heights of Arthur's Seat, Leadhills, and Tinto, above the adjacent plains. He followed the original mode of using a tube sealed at the top, with a paper scale pasted against the side, which he carried to the top of the mountain, where he filled it with mercury; and, inverting it in a basin, he noted the altitude of the suspended column, and repeated the same experiment below; a very rude method, certainly, but no better was practised in England during the succeeding thirty years.

In a small scarce tract, printed in 1688, and bearing the quaint title of Proteus bound with Chains, Sinclair gives some judicious remarks on the variations of the barometer, considered as a weather-glass, and delivers very sound opinions, on the whole, respecting the causes of the chief meteorological phenomena. In a postscript to that piece, he proposes a most efficient and ingenious method of weighing up wrecks from the bottom of the sea. It consisted in employing two large arks, or square wooden boxes, fastened to the sides of the ship, and charged with air carried down to them by a succession of inverted casks, open at the lower end. An arc of a cubical shape, and twenty feet in every dimension, the smallest which he mentions, would, as he computes, have a buoyancy equivalent to 448,000 pounds Troy. It is remarkable that the celebrated Mr Watt always employed this very mode, using a large gazometer, floating in a pond dug in the court of his manufactory, and charged gradually by the action of bellows, for raising the ponderous engines constructed at Soho, and lifting them over his walls into the boats, which were stationed to receive them in the adjacent canal.

In all the computations hitherto made from different altitudes of the barometer, the air was considered as a uniform fluid; no regard being had to the gradual diminution of density which must evidently take place in ascending the atmosphere. To estimate the effect of that gradation, it became requisite previously to determine the actual relation subsisting between the density of the fluid and its elasticity. This was first ascertained in England by Townley, who inferred, from some experiments of Boyle, that the elastic force which the air exerts is exactly proportional to its density. A similar conclusion was about the same time drawn by Mariotte, a French philosopher, from a still better series of experiments. Following out this very simple law, he thought of comparing heights from barometrical observations, by the rules usually employed in constructing tables of logarithms; and had, therefore, obtained some glimpse, no doubt by a sort of conjectural process, of the remarkable result, that the density of the atmosphere decreases in a geometrical progression, corresponding to the elevations taken after an arithmetical one. But seemingly not aware of the importance of the principle at which he was pointing, Mariotte immediately deserted it; and calculating from a repeated bisection of the column of air between the two stations into successive horizontal strata, he contented himself with interpolating the densities according to a harmonic division, which he next abandoned for the simplicity of a series with equal differences. This able experimenter hence only sketched out a mode of investigating the problem of barometrical measurements, without arriving at any very definite or consistent rule of solution.

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1 Sinclair was author of a well-known little book entitled Satan's Invisible World Displayed, which, at a former period, was sold at all the public fairs in the country, and devoured with eagerness and dismay by the Scottish peasantry. In a quarto volume, on Hydrostatics, and the Working of Coal-mine, printed in Holland, and published by subscription in 1672, he digressed so widely from his subject as to insert A True Relation of the Witches of Glenluce. But this was the folly of the age, from which several of the most learned men had not been able to escape. It is painful to observe, that James Gregory, the inventor of the reflecting telescope, who, although endowed with talents of the highest order, appears to have had a keen temper, and to have imbibed an hereditary attachment to royalism and episcopacy, should have stooped to attack an unoffending and less fortunate rival. He wrote a little tract against Sinclair's Hydrostatics, with the title of the Art of Weighing Vanity, and under the thin disguise of Patrick Mather, arch-bendle to the university of St Andrew's. It is a piece full of low scurrility, and memorable only for a very short Latin paper appended to it, containing the series first given to represent the motion of a pendulum in a circular arc. In the British Museum there is a letter of Gregory to Collins, the secretary of the Royal Society, boasting of his project, and soliciting information with which to overwhelm the poor author. But with all his eagerness to hunt down Sinclair, he never touches on the strange episode of the Witches of Glenluce. What a picture of times approaching so near to our own! Sinclair was restored to his chair at the Revolution, and lived a few years longer. He answered Gregory's attack in the same coarse style, charging him with total want of skill in the use of astronomical instruments, though by help of subscriptions he had erected a sort of observatory at St Andrew's. The manuscript of this reply is preserved in the College Library of Glasgow. In 1686 that ingenious and active philosopher Dr Halley resumed the subject, and discovered the law that connects the elevation of the atmosphere with its density; of which he gave a clear demonstration, derived from the well-known properties of the hyperbola referred to its asymptotes. Since the height of the mercury indicates the pressure, and consequently the elasticity of the external air, it must be proportioned likewise to the density. Therefore the breadth of a given mass of air, or the thickness of a stratum which corresponds to a certain portion of the mercurial column, will be inversely as this altitude.

Let O be the centre of a rectangular hyperbola, of which OA and OP are the asymptotes; and conceive the distances OA and OB to represent the heights of the mercury at two stations. The perpendiculars AC and BD, which are reciprocally as OA and OB, must hence express the relative thickness of strata corresponding to equal portions of the barometric scale. Divide AB into a multitude of equal segments, and erect the perpendiculars EM, FL, GK, and HI. The intercluded spaces, from AC to BD, will denote the successive thickness of the series of strata into which the whole mass of air between the two stations is subdivided. Consequently the aggregate or mixtilinear space DBAC, which is proportional to the logarithm of the ratio of OB to OA, will express the difference of atmospheric elevation when the mercurial column mounts from B to A. Taking equal ascents, therefore, in the atmosphere, the corresponding densities must, from the property of the hyperbola, form a decreasing geometrical series.

To apply this elegant theorem, Dr Halley availed himself of the best experiments which had been performed to determine the relative densities of air, water, and mercury. In different trials made near the earth's surface, it was found, when the barometer stood at 29½ inches, that the air is 840, 852, or even 860 times lighter than water. Employing round numbers, therefore, and assuming the specific gravity of mercury to be 13½, he reckoned $800 \times 13\frac{1}{2} \times 30 = 10,800$ inches, or 90 feet, as the altitude of an atmospheric column which, near the level of the sea, would exert a pressure equivalent to that of an inch of mercury. For the co-efficient, which answers to the actual constitution of the atmosphere, Halley should have taken the thirtieth part of $4342945$, the modulus of the common system of logarithms, or $0.0144748$. But he proceeded less directly, having satisfied himself with taking the arithmetical mean between the differences of the logarithms of 29 and 30, and of those of 30 and 31; a compensation of errors which gives $0.0144765$, hardly deviating from the former. Hence he gave this simple analogy for computing the heights of mountains by the barometer:

As the constant number $0.0144765$ is to the difference between the logarithms of the barometric columns at the two stations, so is 900 feet to the elevation required. The result of this operation is evidently the same as if the logarithmic difference had been multiplied by the number 62170; a very tolerable approximation at all seasons for a northern climate, and quite accurate, indeed, if the mass of intervening air had a medium temperature of $46^\circ$ Fahrenheit's scale. Dr Halley supposed that the observations themselves might, from the influence of heat, differ about a fifteenth part between summer and winter. But the thermometer was still so imperfect an instrument, that it could not be applied with confidence in correcting such variations.

The principle which Halley thus investigated might be otherwise derived from a simpler process. Conceive the atmosphere investigated to be divided into a multitude of equally thin horizontal layers, it is evident that each successive stratum would, to the pressure of the superincumbent stratum, add its own weight, which being as its density or elasticity, is therefore proportioned to the collective pressure; and, consequently, those densities must continually increase in going downwards, exactly in the same way, and after a like progression, as money accumulates at compound interest, where a constant portion of the aggregate fund is regularly joined to the capital. Such, in fact, is the distinguishing character of a geometrical progression, that the increase or decrease of each succeeding term is always proportional to the term itself. The logarithmic curve is hence the best adapted for exhibiting the relations which connect the densities with the elevations in the atmosphere; the axis of the curve expressing the elevation, while each ordinate represents the corresponding density of the stratum of air. It being a fundamental property of the logarithmic curve, that every subtangent applied to it has the same length, the exact determination of this in the case of our atmosphere is the only thing wanted for the final solution of the general problem.

Let AB, the absciss of a logarithmic curve, represent a line descending perpendicular through the atmosphere; the ordinates AC and BD will indicate the elasticity of the strata at A and B, and consequently, abating the influence of temperature, their densities. Draw the tangent DT, and form the elemental triangle Ddδ; then BD : BT :: δd : δD, or inversely BD : δd :: BT : δD. But the thickness δD of each accrescent stratum being assumed to be the same, and the subtangent being constant, from the nature of the curve; the ratio of BT to δD is given; therefore the ratio of BD to δd is likewise given, or the increment of the density in the successive strata is proportional to

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1 It appears, however, that he was anticipated by Huygens, who had found, as early as the year 1662, that the altitudes of successive strata of air being taken in arithmetical progression, the corresponding densities will form a geometrical series. This beautiful proposition he never published, indeed, but left it recorded upon one of the blank leaves of a Dutch almanac, containing various other speculations, and now preserved in the library of the university of Leyden.

2 To illustrate still further this comparison; if we reckon the mean altitude of a uniform atmosphere to be 26090 feet, it will follow that, for every descent of 26 feet, the density of the air will increase one thousandth-part. This rate of progression corresponds precisely with the weekly improvement of money invested at 5 per cent. per annum, supposing, for the sake of simplicity, the year to contain only 50 weeks. Since money at the statute interest doubles in 14½ years, so the air has its dilatation doubled at the height of $26 \times 50 \times 14\frac{1}{2}$ or 18466 feet, answering to three miles and a half. the density itself, the leading property of air. The pressure of the thin stratum at B is represented by BDdb, which is equal to the rectangle under BT and δd. Hence, drawing the parallel DE, the collective area ABDC, which represents the weight of the atmosphere between the altitudes A and B, is equal to the rectangle under the subtangent and EC. But AB expresses the difference of the logarithms of the ordinates AC and BD, or the densities at A and B; and consequently this difference multiplied by the length of BT will indicate AB. In relation to our atmosphere at the point of congelation, the numerical value of the subtangent is about 26,000 feet, which is very nearly the product of 60,000 into the modulus of the ordinary logarithms.

Eleven years after Dr Halley had given his rule for barometrical measurements, this philosopher had an opportunity of applying it to discover the height of Snowdon in North Wales. He found that the barometer which stood at 29-9 inches on the sea-shore near Caernarvon, fell a few hours after, when planted on the summit of the mountain, to 26-1 inches, the altitude having been ascertained previously, by a trigonometrical observation, to be 1240 yards.

The year 1687 is memorable as being the date of the first publication of the Principia, which was drawn up chiefly at the urgent request of Halley, from disjointed materials that had lain a considerable time in the author's closet. In that immortal work, Newton resumed the problem of the gradation of atmospheric density, and solved it in that general way which suited his penetrating genius. He demonstrated that, supposing the particles of air, like other bodies, to have their weight or gravitating tendency diminished as the squares of their distances from the centre of the earth, if those distances be taken in harmonic progression, the corresponding densities of the atmosphere will form a geometrical one. But since the diminution of attraction at the greatest height we are able to reach amounts only to the two thousandth part of the whole, this difference is too minute to be adopted in practice; and the simpler law first established by the sagacity of Halley may be deemed sufficiently accurate for every real purpose.

Newton has given a sort of geometrical solution of the problem. But a more precise, and, in this case, a clearer investigation, is obtained by help of the symbols of the integral calculus. Let \( x \) and \( x' \) express the altitudes of two strata of atmosphere, \( y \) and \( y' \) the corresponding densities, \( r \) the radius of the earth; and suppose further, that \( e \) represents the altitude of the equiponderant column which measures the elasticity of the air. Since the density of the air depends on the incumbent pressure, its decrement must evidently be proportional to the weight of each superadded minute stratum, or to the density of this stratum multiplied into its thickness and power of gravitation.

\[ \text{Whence } -\frac{ed}{y} = ydx \left( \frac{r}{r+x} \right)^2, \quad \text{or } -\frac{ed}{y} = \frac{r^2dx}{(r+x)^2}, \]

of which the complete integral is \( e \) Hyp. Log. \( \frac{y'}{y} = \frac{r^2}{r+x} \).

If \( r \) be regarded as indefinitely great in comparison of \( x \), the expression will pass into \( e \) Hyp. Log. \( \frac{y'}{y} = x' - x \), which is only the common formula.

Little seemed wanting, therefore, to complete the practice of the tide of barometrical measurements, but the application of thermometers to correct the results. This instrument, however, advanced slowly to perfection, and more than forty years yet elapsed before it came into current use. Some of the continental philosophers likewise, biased, perhaps, by a secret jealousy of the superiority which England had acquired in science, began to throw out doubts respecting the reality or accuracy of the law of geometrical progression in the atmosphere. Daniel Bernoulli, a man of candour on the whole, as well as ingenuity, but who, with some proneness to speculative reasoning, had unfortunately imbibed many of the prejudices of the Cartesian and Leibnitzian schools, proposed in his capital work, the Hydrodynamica, which appeared in 1736, some vague hypotheses regarding the constitution of the atmosphere, as deduced from certain internal motions attributed to its component strata. The specious results of these calculations led him hastily to deviate from the principle of the geometrical progression of density in the upper regions. In this departure from nice theory he was followed by Cassini and Horrebow, who concluded, from some partial observations they had made, that the barometer, in its indications of atmospheric pressure, is subject to irregularity; and that, near the surface of the earth, it obeys a different law from that which obtains at great elevations. A strong light, however, was thrown upon the subject in 1753 by Bouguer, an able mathematician and very skilful and ingenious observer, who, with other academicians, had been employed for several years in measuring a degree of the meridian along the stupendous ridge of the Andes. From the comparison of more than thirty distinct observations, he deduced a simple and elegant rule for computing heights by means of the barometer. It is this, that the difference between the logarithms of the mercurial columns at the two stations being diminished by one thirtieth part, and the decimal point shifted four places to the right, will express the required elevation in toises. Since the English was to the French foot nearly as fifteen to sixteen, the rule would be accommodated to our measures, and the result expressed in feet, if the logarithmic difference were augmented by the thirtieth part, then multiplied by six, and the decimal point thrown back four places; or, which is the same thing, if that logarithmic difference were multiplied at once by 62,000. But Bouguer imagined, that this rule would not hold exactly in Europe, or in the lower regions of the torrid zone; and to explain the deviation, he had recourse to the forced supposition that the particles of air possess different degrees of elasticity. Lambert, a philosopher of great originality and penetration, afterwards published some excellent remarks on the comparison of barometrical measurements. But no material progress was made till 1755, when M. de Luc of Geneva resumed the subject, and carefully combined experiment with observation. For the space of fifteen years and upwards, he prosecuted his inquiries with diligence and perseverance, aided by the peculiar advantages of local situation, in a city abounding with skilful artists, and seated in the neighbourhood of lofty mountains. The discrepancies which had hitherto created so much embarrassment proceeded mostly from the inattention of observers to the disturbing influence of heat, and particularly its effect in expanding the air, and consequently augmenting the elevation due to a given difference of atmospheric pressure. De Luc's first object was to improve the thermometer of Reaumur, which, though greatly inferior to that of Fahrenheit, had been adopted in France and the adjacent parts of the Continent. Having ascertained that mercury has the valuable property of expanding equably with equal additions of heat, he substituted that metallic fluid for spirit of wine, but retained its arbitrary and inconvenient scale of eighty degrees between the points of freezing and boiling water. He next examined the dilatation of air, at different temperatures, and corrected those results by numerous observations made on the mountains of Savoy, and the mines of the Hartz, in which the barometer was combined with the thermometer. The formula which he thence deduced for the computation of barometrical measurements was, in 1772, published in his *Recherches sur les Modifications de l'Atmosphère*, and seemed to attract, especially in England, a very considerable degree of notice. Dr Maskelyne, the astronomer-royal, adapted it to our system of measures, and De Horsley made annotations and comments on it. But, what was of more importance, other accurate observers, incited by De Luc's example, entered the same field of inquiry, provided with instruments of greater delicacy and much better construction.

In 1775 Sir George Shuckburgh Evelyn visited the Alps, and combined trigonometrical operations with corresponding observations by barometers and thermometers from the hands of Ramsden; and about this time likewise, General Roy not only measured, with instruments made by that excellent artist, some of the principal mountains in Scotland and Wales, but instituted a series of manometrical experiments. It resulted from all these researches that, for each degree on Fahrenheit's scale, mercury expands the 9700th part, and air the 435th part of their respective bulks. It further appeared that the atmosphere has its temperature almost uniformly diminished at equal ascents; and that the logarithmic difference, reckoning as integers the first four digits, expresses in English fathoms the height of an aerial column as cold as the point of congelation. General Roy proposed likewise another correction depending on the enfeebled gravity, and consequently the augmented altitude, of the equiponderant column of atmosphere in the lower latitudes, occasioned by the influence of centrifugal force arising from the earth's rotation. Several years afterwards, Professor Playfair, in a learned paper, printed in the first volume of the *Transactions of the Royal Society of Edinburgh*, examined all the circumstances which can affect barometrical measurements, and discussed each question with the correctness and perspicuity that we might expect from his distinguished abilities. At nearly an equal interval of time the celebrated Laplace resumed the subject in his *Mécanique Céleste*, and brought all the conditions together in a very complicated formula. Such an appearance of extreme accuracy, however, is perhaps to be regarded merely as a theoretical illusion, unsuited and inapplicable to any real state of practice. Biot has since attempted to arrive at a similar conclusion, by setting out *a priori* from some careful experiments on the relative density of air and mercury, performed by him in conjunction with Arago. He thence infers that, in the latitude of Paris, and at the point of congelation, air, under a mercurial pressure of 76 metres, or 29,922 English inches, is 10,463 times lighter than mercury at the temperature of water at its lowest contraction. This would give 26,090 feet for the height of a column of homogeneous fluid, whose pressure is equivalent to the elasticity of the atmosphere. The co-efficient adapted to common logarithms, and adjusted to the force of attraction at the level of the sea, would therefore be 60,148 feet, or 18,334 metres; scarcely differing sensibly from the quantity which Ramond had deduced from a very numerous set of experiments made by him on the Pyrenees. But Biot prefers, as the co-efficient, the number 18,393, answering for an elevation of 1200 metres, or about 4000 feet above the sea, which is not far from the general level of such observations. The formula is hence, in English feet,

$$60,346 \left(1 + \frac{0.02837 \cos 2\phi}{1000}\right) \log_e \frac{H}{h}.$$ Logical traveller, whose object is rather to extend our acquaintance with the altitudes of mountains, than to aim at a superfluous and often illusory precision. The portable instrument, invented by Sir Henry Englefield, and represented in fig. 14, will, on the whole, answer those views. Its cistern is formed of box-wood, sufficiently tight to hold the mercury, without preventing the access and impression of the external air. When this barometer is inverted, the mercury, therefore, subsides very slowly in the tube, which must be firmly suspended in a vertical position. For greater security, the mercury is now put into a leathern bag introduced within the cistern.

A very simple and convenient sort of portable barometer was lately invented in France by that celebrated chemical philosopher M. Gay-Lussac. (See fig. 15 and 16.) It consists of rather a wide siphon tube, filled with mercury, and sealed hermetically at the inverted end, having a very fine capillary hole formed about an inch under this, by nicely directing the flame of a blow-pipe against the side of the glass, and drawing a melted spot of it out to a point. The lower portion of the principal branch has its bore contracted to less than the tenth part of an inch, to prevent the mercurial column from dividing in the act of inverting it. The mercury is boiled as usual, and the tube may be concealed in a walking-stick, or lodged, like the complete mountain barometer, in a cylinder of brass, with movable sliders bearing the divisions of a Vernier at both ends. (See fig. 17.) For greater simplicity, however, the larger divisions might be engraved on the tube itself. This kind of barometer is of ready use, and very little exposed to hazard in carriage. It is commonly held in a reclined or inverted position; but, in making an observation, it must be gently turned back, and kept perpendicular till the mercury descends through the contracted bore, and slowly rises again in the opposite short branch; the scale is noticed at both ends of the incurved column, and the difference of those indications gives its correct altitude.

A modification of the conical barometer, which, in travelling, we have ourselves employed with great ease and advantage, should likewise be mentioned. The principal part of it consists of a small stop-cock made of steel, and represented in fig. 13. A glass tube of 31 or 32 inches long, with a bore of the tenth part of an inch hermetically sealed at the top, and filled with quicksilver, is cemented into the one end of the stop-cock; and into the other end is cemented an open and wider tube, 16 inches or more in length, and having its diameter above the eighth part of an inch. This compound tube is lodged in a walking-stick, divided into inches and tenths through its whole extent, or only at the upper part, if uniform tubes be selected. In making an observation, the cock is turned, and the instrument inverted. The upper column then descends partly into the lower tube, till it becomes shortened to the proper altitude.

We have already stated the principles on which the calculation of barometrical measurements proceeds. But there still are some points, either assumed or overlooked, which may considerably modify the results. It is presumed, that, at equal successive heights, the temperature of the atmosphere decreases uniformly. This property, however, does not hold strictly, and it may be shown, from a comparison of the best observations, that the decrements of heat follow a quicker progression in the higher regions. But we shall soon have another opportunity to examine this subject, and trace out its various consequences.

The humidity of the air also materially affects its elasticity, and the hygrometer should therefore be conjoined with the thermometer in correcting barometrical observations. But nothing satisfactory has yet been done in regard to that subject. The ordinary hygrometers, or rather hygroscopes, are mere toys, and their application to science is altogether hypothetical. A most unphilosophical course has lately been pursued, by multiplying calculations grounded on very loose data, instead of instituting a nice and elaborate train of original experiments.

In the actual state of physical science, it is preposterous, therefore, to affect any high refinement in the formula for computing barometrical measurements. The whole operation may be reduced to a very short and easy process. But the simplicity of the calculation would be still greater, if the centesimal thermometer were generally adopted. It will be sufficiently accurate, till better data are obtained, to assume the expansion of mercury by heat as equal to the 500th part of its bulk for every centesimal degree, while that of air is twenty times greater, being an expansion for each degree of the 250th part of the bulk of this fluid. 1. Correct the length of the mercurial column at the upper station, adding to it the product of its multiplication twice the difference between the degrees on the attached thermometers, the decimal point being shifted four places to the left. 2. Subtract the logarithm of this corrected length from that of the lower column, multiply by six, and move the decimal point four places to the right; the result is the approximate elevation expressed in English feet. 3. Correct this approximate elevation by shifting the decimal point three places back to the right, and multiplying by twice the sum of the degrees on the detached thermometers; this product being now added, will give the true elevation.

If it were judged worth while to make any allowance for the effect of centrifugal force, this will be easily done before the last multiplication takes place, by adding to twice the degrees on the detached thermometers the fifth part of the mean temperature corresponding to the latitude. The mean temperature itself is formed by multiplying the square of the cosine of the latitude by twenty-nine.

In illustration of these rules, we shall subjoin some real examples. General Roy, in the month of August 1775, observed the barometer on Caernarvon Quay, at 30°91 inches, the attached centesimal thermometer indicating 15°7, and the detached 15°6; while, on the peak of Snowdon, the barometer fell to 26°409 inches, and the attached and detached thermometers marked respectively 10°0 and 8°8. Here twice the difference of the attached thermometers is 11°4, and twice the sum of the detached thermometers is 48°8, which becomes 50°8 when augmented by the fifth part of the mean temperature on that parallel. Now, omitting the lower decimals, the first correction is \(0.0264 \times 11.4 = 0.30\), to be added to 26°409. Therefore,

\[ \begin{align*} \log_2 30.091 &= 1.4748486 \\ \log_2 26.439 &= 1.4223450 \\ \text{Difference} &= -0.0525036 \\ \text{Constant multiplier} &= 60000 \\ \text{Approximate height} &= 3368.496 \end{align*} \]

And, for the true height, the correction is \(3.37 \times 50.8 = 171.2\), which gives 3340 for the final result.

We shall take another example from the observations made by Sir George Shuckburgh Evelyn, at the same period, among the mountains of Savoy. This accurate philosopher found the barometer, placed in a cabin near the base of the Mole, and only 672 feet above the surface of the lake of Geneva, to stand at 29°152 inches, while the attached and detached thermometers indicated 16°3 and 17°4; but, in another barometer carried to the summit of that lofty insulated mountain, the mercury sunk to 24°176 inches, the attached and detached thermometers marking 14°4 and 13°4. Therefore, twice the difference of the degrees on the attached was $3^\circ8$, and twice the sum of the degrees on the detached thermometers was $61^\circ6$. Consequently the correction to be applied to the higher column was $0.024 \times 3.8 = 0.09$, which makes it $4.185$. Now,

\[ \begin{align*} \log_{10} 28.152 &= 1.4495092 \\ \log_{10} 24.183 &= 1.3885461 \\ \text{Difference} &= 0.0609631 \\ \text{Constant multiplier} &= 60000 \\ \text{Approximate elevation} &= 3957.786 \end{align*} \]

To correct this approximate elevation, remove the decimal point three places back, and multiply it by $61^\circ6$, increased by $2^\circ9$, the fifth part of the mean temperature, corresponding to the latitude; but $3.96 \times 6.45 = 255.4$, and $3957.8 + 255.4 = 4213$. Hence the summit of the Mole is 4885 feet above the lake of Geneva, or 6083 feet above the level of the Mediterranean Sea.

The last example we shall give is drawn from the observation which Baron Humboldt made among the Andes, near the summit of Chimborazo, the highest spot ever approached by man. This celebrated traveller found there, that the barometer fell to 14,850 English inches; the attached thermometer in the tent being at $10^\circ$, and the detached in open air being $1^\circ$ under zero. But the same barometer, carried down to the shore of the Pacific Ocean, rose exactly to thirty inches, while both the attached and detached thermometers stood at $25^\circ3$. Consequently the correction to be applied to the upper column is $-0.015 \times 30.6 = -0.45$. Wherefore,

\[ \begin{align*} \log_{10} 30.000 &= 1.4771213 \\ \log_{10} 14.895 &= 1.1730405 \\ \text{Difference} &= 0.3040808 \\ \text{Constant multiplier} &= 60000 \\ \text{Approximate elevation} &= 18244.848 \end{align*} \]

Now, the difference of the detached thermometers, or $2^\circ9$, being doubled and further increased by $5^\circ8$, the fifth part of the mean temperature at the equator, makes $59^\circ6$; and the final correction to be applied is therefore $18.24 \times 59.6 = 1087$, which gives 19,332 feet for the true elevation observed, or 2140 feet below the summit of Chimborazo.

These calculations are performed by the help of logarithms. It is desirable, however, to approximate at least to barometrical measurements without such aid. A very simple rule for this object has been given by Professor Leslie in his *Elements of Geometry*. Since $\log_{10} \frac{a}{b} = 2M \left( \frac{a-b}{a+b} + \frac{1}{3} \left( \frac{a-b}{a+b} \right)^3 + \frac{1}{5} \left( \frac{a-b}{a+b} \right)^5 \right)$, where $M$ denotes the modulus of the logarithmic system; when $a$ approaches to $b$, the lower terms may be rejected without sensible error, or $\log_{10} \frac{a}{b} = 2M \left( \frac{a-b}{a+b} \right)$, very nearly.

Therefore, in reference to our atmosphere, the modulus is expressed by the equiponderant column of homogeneous fluid, or $60,000 \times 4342945 = 26,058$ feet, or only $26,000$ in round numbers; whence, as the sum of the mercurial columns is to their difference, so is the constant number $52,000$ feet to the approximate height. Let General Roy's observation on Snowdon be resumed as an example: the analogy is $30.091 + 26.439 = 30.091 - 26.439$, or $56.550 : 3.652 :: 52,000 : 3,359$, the approximate elevation, differing very little from the logarithmic result.

This mode of calculation may be deemed sufficiently accurate for determining any altitude that exceeds not 5000 feet. But it will extend to greater elevations if the second term of the series be likewise taken, which is done by striking off three figures, and cubing the half of this number. Thus, resuming the mensuration of Chimborazo, $44.895 : 15.105 :: 52,000 : 17,496$, and $(8.75)^2 = 670$, making together 18,166 for a nearer approximation.

The calculation of barometrical measurements, including the corrections required, is rendered most easy and expeditious by means of a sliding rule made by Mr Cary, optician in London. This small instrument should always go along with mountain barometers; and it will be found a very agreeable companion to every geological traveller.

But portable barometers, in spite of every precaution, are yet so liable to be broken or deranged, that other auxiliary methods are desirable for ascertaining distant elevations. In this view, the variation of the boiling point of water was proposed by Fahrenheit as far back as the year 1724; the idea having occurred to him, as it had done before to Amontons, while engaged with experiments to perfect his thermometer. Little regard, however, seems to have been paid to the suggestion till De Luc and Saussure made a series of observations on the heat of ebullition at different elevations above the surface. About thirty years since, Cavallo attempted to revive the scheme of Fahrenheit, but experienced much difficulty in preventing the irregular starts of the thermometer plunged in boiling water. The best and surest way of examining the heat of ebullition, is to suspend the bulb of the thermometer in the confined steam as it rises from the water.

The heat at which water boils, or passes into the form of steam, depends on the weight of the superincumbent atmosphere. By diminishing this pressure, the point of boiling ebullition is always lowered. It appears that, while the atmospheric pressure decreases exactly, or at least extremely nearly, in a geometrical progression, it being found that every time such pressure is reduced to one half, the temperature of boiling water suffers a regular diminution of about eighteen centesimal degrees. This beautiful relation assimilates with the law which connects the density and elevation of the successive strata of the atmosphere. The interval noticed between the boiling points at two distinct stations must be proportional to their difference of altitude above the level of the sea. We have, therefore, only to determine the co-efficient or constant multiplier, which may be discovered either from an experiment under the rarefied receiver of an air-pump, or from an actual observation performed at the bottom and on the top of some lofty mountain. We shall prefer at present the observation made by Saussure on the summit of Mont Blanc. This diligent philosopher found, by means of a very delicate thermometer constructed on purpose, that water which boiled at $101^\circ52$ in the plain below when the barometer stood at $30.534$ English inches, boiled at $86^\circ24$ on the top of that mountain, while the barometer had sunk to $17.136$. Wherefore the distance between the points of ebullition, or $15.38$ centesimal degrees, must correspond to an approximate elevation of $15,050$ feet, which gives $978\frac{1}{2}$ feet of ascent for each degree, supposing the mean temperature of the atmospheric column to be that of congelation. But it will be more convenient to assume $1000$ for the constant multiplier, which corresponds to the temperature of $5^\circ2$.

To reduce this very simple result into practice, it would be requisite to have a thermometer with a fine capillary bore, and nicely constructed, the stem six or eight inches long, and bearing ten or a few more degrees from the boiling point; these degrees to be divided into twenty or perhaps fifty equal parts engraved on the tube, which should be rather thick, and terminating in a bulb of about half an inch diameter. This thermometer, being fitted with a brass ring two inches above the bulb, should screw into the narrow neck of a small copper flask, which holds some water, but has a hole perforated near the top for allowing the steam to escape. The water may be made to boil by the application of a lamp. The difference between the indications of the thermometers at the two stations being multiplied by a thousand feet, will give the elevation corresponding to a temperature of $51^\circ$. The correction for the actual mean temperature can easily be applied. If a more correct co-efficient be afterwards determined, the same thousand, retained as a multiplier, may easily be adapted to another temperature.

This method of measuring elevations on the surface of the globe is, therefore, capable of great improvement, and might be employed with advantage in a variety of cases where observations with the barometer are not easily obtained. Its application would be most important to physical geography, in ascertaining the capital points for tracing the outline of the profile or vertical section of any country. The common maps, which exhibit mere superficial extension, are quite insufficient to represent the great features of nature, since the climate and productions of any place depend as much on its elevation above the sea as its latitude. Scientific travellers have accordingly turned their attention of late years to the framing of vertical sections. As a specimen, we give, in fig. 22, from Humboldt's Geography of Plants, a section across the American continent, one of the best and most interesting that has yet appeared. It consists, in fact, of four combined sections, traversing through an extent of 425 miles. The line begins at Acapulco, on the shore of the Pacific Ocean, and runs 195 miles, about a point of the compass towards the east of north, to the city of Mexico; then 80 miles, a point to the south of east, to La Puebla de los Angeles; again it holds a north-east direction of 70 miles, to the Cruz Blanca; and finally bends 80 miles east by south, to Vera Cruz, on the coast of the Atlantic. A scale of altitudes is annexed, which shows the vast elevation of the table-land of Mexico. An attempt is likewise made in this profile to give some idea of the geological structure of the external crust. Limestone is represented by straight lines slightly inclined from the horizontal position; Basalt, by straight lines slightly reclined from the perpendicular; Porphyry, by waved lines somewhat reclined; Granite, by confused hatches; Amygdaloid, by confused points.

By this mode of distant levelling, a very interesting discovery, in another quarter of our globe, was made by Engelhardt and Parrot, two Prussian travellers. They proceeded, on the 13th July 1814, from the mouth of the Kuban, at the island of Taman, on the Black Sea; and, examining carefully every day the state of the barometer, they advanced with fifty-one observations, the distance of 990 wersts, or 711 English miles, to the mouth of the Terek, on the margin of the Caspian Sea. Similar observations were repeated and multiplied on their return. From a diligent comparison of the whole, it follows that the Caspian is 334 English feet below the level of the Black Sea. That the Caspian really occupies a lower level than the Ocean, had been suspected before, from a comparison of some registers of barometers kept at St. Petersburg, and on the borders of that inland sea; but the last observation places the question beyond all doubt. It further appears, that within 250 wersts, or 189 miles, of the Caspian, the country is already depressed to the level of the Ocean, leaving, therefore, an immense basin, from which the waters are supposed to have retired by a subterranean percolation.

If the same plan of barometrical measurements had been followed by the adventurous explorers of the African continent, and the altitudes of the central lakes ascertained, the question regarding the course of the Niger would have been much sooner settled, and much vague and unsatisfactory discussion avoided. Even thermometrical observations of the temperature of springs, or of the ground at moderate depths, would have furnished an approximation to those elevations sufficiently near for solving that great problem in geography on scientific principles alone.

The spirit of modern enterprise has more lately carried barometers to the remotest and loftiest stations on the surface of our globe. It has been thus ascertained that some of the mountains in the equatorial parts of America attain the stupendous altitude of 25,000 feet, while the great chain of Upper India seems to rear its vast summit about 2000 feet still higher. See Climate, Hygrometry, and Meteorology.