and by him they were made known to various other British mathematicians. He likewise entered into a correspondence with Collins and Oldenburg, and by them was induced to write several long letters to Mr. Leibnitz, in which he gave an historical detail of the way that he was led to some of his most considerable discoveries. All these letters were afterwards published in the Commercium Epistolicum. The correspondence also between James Gregory and Collins, published in the same book, throws considerable light upon the order and time of Newton's mathematical discoveries. One of his first discoveries struck him while perusing Wallis's Arithmetic of Infinites, about the year 1663. Wallis had shown the method of finding the quadrature of all curves, the ordinates of which are expressed by (12)m, x being the abscissa, supposing m a whole number, either positive, or negative, or zero; and that when m was respectively 0, 1, 2, 3, 4, &c., the areas corresponding to the abscissa x were respectively x; x x3 ; x - 2x + 3x2 + 4x3 - 4x; &c.; and be showed, that if a number could be interpolated between x and x- in the second series, corresponding to the interpolation of in the first series between 0 and 1, that this number would represent the quadrature of the circle. But Wallis could not succeed in making this interpolation; it was left for one of the first steps of Newton, in his mathematical career. Newton arranged the terms of the second series given above, under each other in order, and examined them as follows:

x

-

x;

On considering this table, Newton observed, that the first terms are all x; that the signs are alternately positive and negative; that the powers of x increase by the odd numbers; that the coefficient of the first term is 1; that the co-efficient of all the other terms are fractions; that the denominators of these fractions are always the indices of x, in the respective terms; that the numerators in the second terms are the ordinary numbers; in the third terms, the triangular numbers in the fourth terms, the pyramidal numbers; &c. These observations made him master of the laws that regulated the whole of the series. Hence he concluded, that having to develope in general (1 x2), the series of numerators for the respective fractions in the different terms must be 1; m;

which represent the natural, triangular, and pyramidal numbers. Now this will hold good whether m be a whole number or a fraction. In the case which occasioned the investigation, namely, x2); m =, and, consequently, the numerators deduced from the preceding formulas are 1, T, &c. These, multiplied into the terms of the series, namely, x

(1

x3

+

5

1

X5 8.5

a7

7

x9

+ &c., give us the following series: x

5

16.737

128.99, &c., a series which ob

viously represents the area of the circular segment, corresponding to the abscissa x. This investigation led him likewise to the discovery of the binomial theorem, so celebrated in algeb a, and of so much importance in an infinite number of investigations.

Newton had already made these discoveries, and many others, when the Logarithmotechnia of Mercator was published; which contains only a particular case of the theory just explained. But, from an excess of modesty and of diffidence, he made no attempt to publish his discoveries, expressing his conviction that mathematicians would discover them all before he was of an age sufficiently mature to appear, with propriety, before the mathematical world. But Dr. Barrow having contracted an acquaintance with him soon after, speedily understood his value, and exhorted him not to conceal so many treasures from men of science: he even prevailed upon him to allow him to transmit to some of his friends in London a paper containing a summary view of some of his discoveries. This paper was afterwards published under the title of Analysis per Equationes Numero Terminorum Infinitas. Besides the method of extracting the roots of all equations, and of reducing fractional and irrational expressions into infinite series, it contains the application of all these discoveries to the quadrature, and the rectification of curves; together with different series for the circle and hyperbola. He does not confine himself to geometrical curves, but gives some examples of the quadrature of mechanical curves. he speaks of a method of tangents, of which he was in possession, in which he was not stopped by surd quantities, and which applied equally well to mechanical and geometrical curves. Finally, we find in this extraordinary paper the method of fluctions and of fluents, explained and demonstrated with sufficient clearness; from which it follows, irresistibly, that before that period he was in possession of that admirable calculus: for the editors of this paper, which was published in the Commercium Epistolicum, attest that it was faithfully taken from the copy which Collus had transcribed, from the manuscript sent by Barrow. At the request of Dr. Barrow, he drew up a full account of this method,

which was only described in the first tract with great conciseness, This new work he entitled Methodus Fluxionum, et Serierum Infinitarum. This last book he meant to publish at the end of an English Translation of the Algebra of Kiuckuysens, which he had enriched with notes. But, in consequence of the disagreeable disputes into which he had been dragged, by his discoveries respecting the different refrangibility of the rays of light, he altered his intention, and the treatise, to the great injury of mathematics, and ultimately, likewise, to the diminution of his own peace, lay unpublished till after his death.

About the time that this paper of Newton's was sent to London, or about the year 1668, James Gregory published his Exercitationes, a book which contained several important facts connected with the discoveries which Newton had made. In particular there is a new demonstration of Mercator's Series for the Hyperbola. Collins communicated Newton's discoveries to various mathematicians, and among others to Gregory. He first sent him Newton's Series for the Circle, concerning the accuracy of which Gregory at first had his doubts; but he soon discovered his mistake, and by pondering over the subject for about a year, there appears sufficient evidence from his letters in the Commercium Epistolicum, that he divined Newton's method, and consequently had the merit of discovering the fluctionary calculus at least in part. But he declined publishing any thing on the subject, as he states in one of his letters, that he might not interfere with the rights of the original inventor.

(To be continued.)

ARTICLE II.

Observations on the Quantity of Carbonic Acid Gas emitted from the Lungs during Respiration, at different Times, and under different Circumstances. By Wm. Prout, M.D. Of the College of Physicians, &c,

(With two Plates.)

Ir was discovered by some of the earliest experimentalists on respiration, that the quantity of oxygen gas consumed, and of carbonic acid gas formed, during that act, varied very considerably in the same individual, under different states of the system. "The circumstance," says Dr. Bostock, "was first noticed by Dr. Crauford, and afterwards more fully investigated by M. Jurine, of Geneva, and M. Lavoisier, that the respiration of the same animal in different states of the system, and under the

operation of different external circumstances, affects the air in very different degrees. The circumstances which have been discovered to influence the chemical effects of respiration are, the temperature of the air respired, the degree of muscular exertion, the state of the digestive organs, and the condition of the system as affected by fever." Dr. B. continues: "It is highly probable that other circumstances will be discovered by multiplying and varying our experiments upon the living body," and informs us, that "these different affections of respiration will undergo a farther examination in the third part of his essay. ** Now this, I believe, has never been published; whether, therefore, he, or any other person, has anticipitated me in what I am about to offer, I am unable to determine: if so, my experiments will at least have the effect of corroborating theirs; and if not, their interesting results may possibly induce some one to repeat them, and thus either confirm their accuracy, or point

out their errors.

Mr. Brande, also, in a paper on respiration, after having noticed the above circumstances in a general way, says, "the proportion, however, (of carbonic acid) varies in the same individual during the 24 hours, for I have found the quantity of carbonic acid gas emitted from my own lungs to be rather less in the morning than towards the evening; but this also varies in different people." It may also be mentioned, that Messrs. Allan and Pepys, in their excellent paper on this function, found the quantity formed by the same animal during sleep to be less than when waking. I

Such then, with perhaps a few other general facts of a similar kind, constituted the whole of our knowledge (or rather of mine) respecting this most important part of the phenomena of respiration, and this, together with some other reasons which will be shortly known to the public, induced me to think of undertaking a set of experiments with the hope of throwing some light upon the subject, and, if possible, to find out the laws which it obeyed. With this view, having contrived a simple apparatus, by means of which I could easily, and with considerable accuracy, analyze the respired air, I put myself upon a sort of regimen, which consisted in keeping myself as nearly as possible in the same state in every respect, and thus commenced the arduous task. To this plan I adhered as nearly as circumstances would permit for upwards of three weeks, making the experiments every hour, and sometimes oftener, during the day, and occasionally during the night also. Now the results obtained from this great mass of evidence, amounting

Essay on Respiration, p. 78.

+ Nich. Journal, vol. xi. Phil. Trans. 1809. Nich. Jour. vol. xxv.

to many hundreds of experiments, were generally consistent, and may be comprised under the two following laws:

Law I.-The quantity of oxygen gas consumed, and consequently of carbonic acid gas formed, during respiration, is not uniformly the same during the 24 hours, but is always greater at one and the same part of the day than at any other, that is to say, its maximum occurs between 10 a. m. and 2 p. m., or generally between 11 a. m. and 1 p. m.; and its minimum commences about 8 30 p.m., and continues nearly uniform till about 3 30' a. m.

Law II.-Whenever the quantity of oxygen gas consumed, and consequently of carbonic acid gas formed, has been by any cause increased or raised above the natural standard of the period, it is subsequently as much decreased or depressed below that standard, and vice versâ.

Illustration of Law 1.-This law is subject to some remarkable variations, though I have never met with an exception to it. In all my experiments there has been constantly a greater quantity of carbonic acid gas given off in the middle of the day than at any other period of it. From what Mr. Brande advanced in the paper above alluded to, I was indeed prepared to meet with the reverse of this; and for some time felt inclined to suspect the accuracy of my experiments, till, by varying them in almost every possible manner, and with the same results, I could no longer resist their united evidence.

Generally, the degree and order of these variations are the following. The quantity of carbonic acid gas, which has remained stationary during the night at 3.30 per cent.,* its minimum, about 3 30 a. m.,† suddenly begins to increase, at first slowly, and afterwards more rapidly, till about noon, when it is usually as high as 4.10 per cent., or its maximum; from this point, however, it almost immediately begins to sink, at first rather quickly, and then more slowly, till about 8" 30′ p.m.,† by which time it generally arrives again at its minimum, 3.30 per cent., when it remains stationary, as before observed, till the morning. Hence the quantity given off in the middle of the day, when it is at its maximum, exceeds that given off in the night, when it is at its minimum, by about of the whole. The mean quantity given off in the 24 hours is 3.45 per cent. (See Table 1.)

By this is meant that for every 100 cubic inches of air inspired, 3.30 cubic Inches of oxygen gas are consumed, and consequently of carbonic acid gas formed. The same is to be understood of all the subsequent numbers.

+ That is, at the beginning and end of twilight. Many circumstances have occurred to induce me to believe that the presence and absence of the sun alone regulate these variations. Future observations, however, must decide this Curious question.