Dr. Black's speculations respecting heat have had such an effect upon the progress of the science, that it would be unpardonable not to notice them here. A very good account of them will be found in the first volume of his lectures. Indeed, his lectures on heat constituted the most finished, and by far the most valuable, part of his course. It is well known that the freezing point of water is 32°, that whenever the thermometer sinks below 32° water begins to freeze, and whenever it rises above 32° ice and snow begin to melt. At the first view of the subject one would be disposed to expect that as soon as the thermometer sinks below 32° the whole water would immediately become ice, and that when it rises above 32° the ice would be as speedily converted into water; but every body knows that these speedy changes never take place. In cold weather a crust of ice is formed upon the surface of rivers and lakes; and if the cold continue, this crust becomes gradually thicker. But unless the water be very shallow, it is very seldom or never totally converted into ice. The warm weather returns while a considerable portion of the water of the lake is still unfrozen. We remark the same slowness in the conversion of ice into water. When snow is accumulated in great quantities in mountainous countries, it resists the united action of the sun and the wind for weeks, or even months. It is always melting, indeed, but it melts very slowly; and in some cases the cold weather returns again before the liquefaction is completed. Such were the facts which had been obvious to all the world from the beginning. Dr. Black was the first person who examined them closely and endeavoured to explain them. According to him, water is a compound of two substances-ice and heat. It cannot freeze or be converted into ice till it has parted with its heat; and as the heat makes its escape but slowly, the water freezes but slowly. Ice, on the other hand, can only be converted into water by combining with a certain quantity of heat; and as this combination takes place but slowly, the ice melts but slowly. This view of the subject Dr. Black confirmed by simple but satisfactory experiments. The heat which thus renders water fluid he called latent heat, because its presence is not indicated by the thermometer. He showed that the latent heat of water is 140°. He ascertained likewise that fluidity in all cases is owing to the combination of latent heat with the body becoming fluid.

It is well known that water and other liquids, when exposed to heat, increase in temperature till they become boiling hot, but after that their temperature remains stationary. They gradually indeed boil away, and are converted into steam or vapour, an elastic fluid possessing many of the properties of air; with this difference, that when exposed to the action of cold it is again converted into the very liquid from which it was originally produced. Dr. Black showed that vapour or steam is a combination of the liquid from which it was produced and latent heat. The latent heat of the vapour of water or steam he found about 950°. It was this doctrine

respecting the nature of steam that led Mr. Watt to his great improvements on the steam-engine-improvements which have been of incalculable benefit to the manufactures of Great Britain.

Dr. Black was the first person who pointed out that every substance is possessed of a peculiar specific heat, or that different bodies have different capacities for heat. This subject was afterwards further investigated by Dr. Irvine, of Glasgow, and by Mr. Wilcke, of Stockholm.

A very short paper by Dr. Black was published in the 65th volume of the Philosophical Transactions, for the year 1775, giving an account of some experiments showing that recently boiled water begins to freeze more speedily than water that has not been boiled. He found that if the unboiled water be continually stirred, it begins to freeze as soon as the boiled water. He gives the following explanation of the phenomenon. Water by boiling is deprived of a portion of air. When exposed to the atmosphere it begins to absorb this air, and continues to do so till it has recovered its original quantity. This absorption produces a disturbance in the water, not indeed sufficient to be perceived by the eye, but sufficient to prevent it from becoming colder than 32°, without beginning to freeze.

The only other paper written by Dr. Black was published in the second volume of the Transactions of the Royal Society of Edinburgh. It is an analysis of the Geyzer and Rikum springs in Iceland. A quantity of the water of these springs was brought from Iceland by Sir John Thomas Stanley, and sent to Dr. Black. This paper may be taken as a model of the proper manner of examining mineral waters. The following were the constituents found in 10,000 grains of each of these waters :—

On the Doctrine of Fluxions. By Alexander Christison, Esq. Professor of Humanity in the University of Edinburgh.

MY DEAR SIR,

(To Dr. Thomson.)

Ir is very much to be regretted that many students at the universities of North Britain acquire no knowledge of fluxions. They

seem to think it extremely difficult to obtain even the first principles of that important branch of mathematical science. In what follows, (which, if you think proper, you may insert in the Annals of Philosophy,) I intend to solve inductively the fluxional problem as extensively as Newton has demonstrated it in the second lemma of the second book; then to demonstrate that problem rigorously in the manner of the ancients, independently of infinitesimals, of motion, or of vanishing quantities; and, lastly, to subjoin some observations.

I am, my dear Sir, yours faithfully,

Edinburgh, March 20, 1815.

Of Fluxions.

ALEX. CHRISTISON.

In consequence of repeated trials, I have long thought that a boy, duly prepared, passes from common algebra into fluxions as easily as he does from multiplication into division.

In solving the fluxional problem three things are to be distinguished-the conception, the notation, the demonstration.

B

1. With regard to the conception, I shall, in order to be easily understood, proceed as if I were questioning a learner. Ask him thus-If a straight line, A D, fig. 1, move parallel to itself at right angles along DC, blackening the parallelogram AF, whose side AE or BF is 5; and reddening the parallelogram E C, whose side ED or FC is unit; at what rate does it always blacken the one parallelogram and redden the other? He will answer-As 5 to 1. The conception is much aided, at first, by his imagining that the two parallelograms are generated of D different colours.

E

Fig. 1.

F

Ask him now thus-If a straight line move parallel to itself, at

Fig. 2.

right angles along E D, fig. 2, so that it can generate only the parallelogram A D, whose side AE is unit, and the triangle ABC half a square, while all the rest of the space is covered; at what rate does it, at HF: = 6, generate the triangle and the parallelogram? He will answer-As 5 to 1'; or as 5 x 11 x 1; that is, as the area of the parallelogram H M is to the area of the parallelogram GL; but he will probably add, that the instant before the rate was less, and that the instant after it will be greater. He may be told, that it is not the rate the instant before, nor the instant after,

5

G

M

E

FL

which is wanted. He will now understand that, if the line FH were to proceed for ever of the same length, it would generate two parallelograms which would have always to one another the rate of 5 to 1. If, therefore, x represent the base or the perpendicular of the triangle, the rate will be as x to 1.

2. With regard to the notation, if x and 1 be both multiplied by any quantity whatever, their rate will not be changed; instead of to 1, therefore, we may employ to 1, being any line more than nothing, and less than infinite: x, then, is the fluxion of the triangle, and 1 & the corresponding fluxion of the parallelogram; and as the triangle is the half of a square, the fluxion of a square whose side is a is to the fluxion of a parallelogram whose side is L

as 2 x x 1 x.

If now the following series of fluents be set down, the learner will easily continue the fluxions.

A learner who sees in the series of the fluxions above the two laws of the three first terms, that of the numeral coefficients, and that of the letters, will be able to continue the series to any length, and to give the general expression n as the fluxion of x"; because he observes that the numeral coefficients increase by unity, and that there are as many letters in the fluxion as in the corresponding fluent, with the last letter always dotted.

If the result be expressed in words, we have the following rule for finding the fluxion of any power of a variable quantity.

Multiply the fluxion of the root by the exponent of the power, and the product by that power of the same root whose exponent is less by unity than the given exponent.

x; of xTM y" is my” xTM−1 y vary, by considering first variable; of

= x y

y"

16

772

¿; of x-is

n

¿ + nx” y"-1j, when both x and

x as variable, and then y as also m y" xm -1 x - is — n xTM y”—1 j

From all that has been said, the fluxional or differential calculus may, in the case of one variable quantity, be defined a method for finding the rate of change in a quantity, and its dependance or function. Thus the rate of change in t and its function a", is as *: n x-1ż, or as 1 : nx”~'.

-1

As this is not a treatise, but a short essay, I say nothing of second fluxions, which bear the same relation to first fluxions that first

fluxions bear to their fluents; nor of exponential and logarithmic quantities; nor of the arithmetic of sines; but I refer for information to some of the authors afterwards mentioned.

3. With regard to the demonstration, I think that the view already exhibited leaves no doubt in the mind of the learner; but a rigorous demonstration should be given, in order to enable him to Teply to every objection.

Newton's second lemma of the second book seems to afford a demonstration that, while it is brief and comprehensive, is convincing, if the reduction to absurdity by Robins, vol. ii.; or by Hales, in the Logarithmic Writers, vol. v. p. 133 and 134; be subjoined to Newton's case first; if the process from particular to general be admitted in his case third; and if the momentum be admitted instead of the fluxion.

In a department of science so important and extensive as fluxions, the demonstrations of various authors should, I think, be studied by the learner; such as those of Simpson, Maclaurin (though extremely tedious), Euler, l'Huilier, Bossut, Vince, Dealtry, Lacroix, and Lagrange.

Any function of a variable quantity may be represented by the ordinate of a curve of which x is the abscissa. Let y=x" be a function of x, and let x become x + i; then y=x will become y1 = x + i}" = x2 + n x* ~ ' ¿ + subtract the first equation from the second, and divide both sides by

i, we shall have ? —

n. n 1 x7-2

1.2

y' -y

n.n 1

1.2

2

i+ &c. If we

i + &c. Now

it is evident that i, and consequently yy, which depends on i, may be so diminished that n x may differ less from n x-1 + i+ &c. than by any assigned quantity how small

1.2

soever; and when in 3, i and y'

y vanish,

seems equal to n x-1. But this conclusion, says Lagrange, 66 presents no

idea,"

FG
EF

Prop. is equal to n x', fig. 3. Construct the figure in

which T G is the tangent. Let