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ones deduced therefrom by Mr. Landen and Mr. Clarke... We would not be understood here to depreciate general forms, which, in compounded cafes, are eligible and useful; what we mean to infinuate is, that there may be bounds beyond which they may cease to be so. Thus, whoever attentively considers Mr. Simpson's propositions above-mentioned will see, that if the series contained in one or more of them, be added, fubducted, multiplied, or otherwise compounded with others, we shall obtain other forms still more general and more complex, and fo on ad infinitum. There must therefore be some limit or other, beyond which it may not be eligible to compound these things. For if such forms, as that given by Mr. Landen at the end of his Obfervations, be enough to affright an old soldier, what shall we say to those that are ten times, nay a hundred or a thousand times more complex. In real practice, we seldom meet with any very compound ones, agreeing at the same time with a preassigned general form; those that occur commonly require some previous reduction or transformation to fit them to the formula. Of this Mr. Simpson (vide p. 99 of his Mathematical Differtations) was well aware, and exemplifies accordingly how needful it is (for avoiding trouble) firit to reduce every series to the most commodious form, before we fet about to determine its vaJue. And we will venture to add, that when it is so reduced, it often appear, at fight, bow without the use of complex forms, it may be farther reduced to very simple ones, whose fums may be immediately apparent. His concluding example at p. 98, of his Differtations, might furnish an instance; but that at p. 85 fupplies one still more remarkable : where from the sum of the series % +

+ + + &c. being given = S, he pro5 9 13 у

6 y 3 10 y. poses to find that of + +

1.5.13

5.9.17 9.13.21 13.17.25 + &c. But in order to adapt this to his forms, he directs to put z* = j, and then shews that the series itself will thus be

4.8
8.12%

12.1628 changed to ***:

+

+ &c. 32 1.5.13 5.9.17 9.13.21 which now agrees with his forms. But it is also now very plain, when it is so transformed, how its sum may readily be found without them. ' For the relation of the factors in the numerators, so as to have constant differences to the same respective ones in order in the denominators, is eafily discovered, perhaps as easily and soon as we could, by comparing the factors of the terms, be sure that the series was adapted to any general formula. Thus then the last series evidently becomes (omitting

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for the sake of brevity the general multiplier :) this following,
5-1.5+3.9-1.9+3.2* 13-1.13+3.28
1.5.13
5.9.17

9.13.21
5+3 5+3 9+3.z* 9+3.z+ 13+ 3.28
+

+ 1.13 1.5.13 5.17 5.9.17

9.21
13+ 3.2

4
8 z*

1228
&c. =

+
9:13.21
1.13 5.17 9.21

1.13 32+

3 32+ &c.

&c, = 5.17 9.21

1.5.13 5.9.17 9.13.21
21-6. zs

3
+
+

&c.
1.13
5.17
9.21

4

1.13 S

27 +

X:
5.13 5.17
9.17 9.21

13.21
&c.; +

3

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5
21

5 13

s &c. ; =

15

X:
9

13
21

482

32
S
3
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; agreeing with 9z+ 52

32 Mr. Simpson's conclusion. And here without recurring at all to any general form, we have put down every figure that we actually made use of in the operation; and all the transformations we have here given are far more evident than that which Mr. Simpson found necessary, in order to be able to reduce the original series within the rank comprehended under his general for. mula. Our limits, however, will not allow us to enlarge further on this subject.

Mr. Claike is indeed very severe upon his antagonist; bringing charges which, if true, would make Mr. Landen appear very little indeed. For he says, at p. 53d of his Supplement .... “And now, gentle Reader, for your farther acquaintance with men, manners, and things, I shall present you (by way of conclufion) with the translation of a few pages from the AEla Eruditorum of Leipzig, for September 1762 (p. 458.); and when you have well digefted, and strictly compared it with that gentleman's second Memoir (p. 23, &c.), particularly the oth and roth Articles thereof, I would advise you to annihilate those wonderful insignia with which that memoir is so profusely embellithed; not only as a just sacrifice to the manes of J. F. De Tuschis de Fagnano and his venerable father, but that the scurriloūs and ill-natured may not thereby take occasion to say, that a British Mathematician had ascribed the honour of a discovery so himself, which he had “ bafely pilfered" from a foreigner'.

He

He then puts down from the above-named Aets, the demonstration of a theorem, containing some properties given by Mr. Landen in his second Memoir, who, in answer, fays, in the poft script of his Appendix. . . * With respect to what may have been discovered by I. F. De Tuschis de Fagnano and his father, I can, with the stricteft regard to truth, aver, that I had never, by any means whatever, seen (or heard of) any of their Works before the publication of Mr. Clarke's

. Supplement: nor have I seen any of the Asta Eruditorum, or Diarium Eruditorum Italia, or the Works printed at Pesaro, mentioned in that pamphlet.-Its author, therefore, may take fhame to himfelf for the illiberality of his refeciions !--If I have proved his want of skill, be himfelf has proved his want of candour'.... And indeed we see no reason for not believing Mr. Landen, his method of investigating these properties being wholly different from what is tranrlated by Mr. C, and bearing all the marks, or internal evidence, of originality.

Abitrufe mathematics are now cultivated by able hands, in so many different nations and languages, that few private persons have the opportunity of seeing all that is done on these subjects. Mr. Clarke seems to us to have an advantage, which those that live in retired situations can rarely possels, namely, that of confulting a good public library." If then he would have taken the trouble, with any tolerable share of candour and impartiality, to have given a short account of the invention and improvement of each formula, ałcribed to their proper owners; it would have been an honour to himself, and an acceptable piece of service to the Public; as our English mathematicians especially are Thamefully deficient in particular acknowlegements,

T

ART. III. An Introduction to Natural Philosophy. Illuftrared with Copper-plates. By William Nicholson. 8vo. 2 Vols.

12 S. Boards. Johnson. 1782. HIS treatise, the Author tells us, is intended to give a

clear account of the present state of Natural Philosophy to such as poflefs very little mathematical knowledge. That his grand object, throughout, has been to relieve the memory, and allift the understanding by conciseness, and illustrative arrangement. And notwithstanding the nature of the undertaking unavoidably required a deviation from those beautiful and general principles, which are obtained by strict mathematical reaioning, yet it is presumed, he says, that the student will find nothing in this treatise which he will be under the necessity of unlearning, when he attempts the perufal of those books, to which this is offered as an introduction. From these short extracts from the Author's own account of this performance, it will be

seen,

seen, that those branches of Physics, which include the nature and laws of motion, mechanics, hydraulics, optics, and astronomy, can be but cursorilġ touched upon in this performance; and indeed this part was the least wanted, because we have already so many excellent elucidations of these branches... What seemed chiefly wanting now, was something in an elementary form, adapted to the capacity of beginners, concerning the great discoveries and improvements made by Dr. Priestley, and others, in Aerial Physics, Chemistry, and Electricity. On the two first of these subjects Mr. Nicholson has been particularly explicit : and as a specimen of his method of proceeding, we shall, for the entertainment of our young philosophical Readers, select his account of phlogiston, vol. ii. p. 160,

• The element, phlogiston, is by many chemists termed fire; and in order to distinguish it from heat, which they likewise call fire, the phlogiston is called combined fire, while heat is denoted by the term Elementary Fire. But this manner of expresfion presupposes a certain theory of fire, which is by no means established. We are ignorant whether heat be a mode or a subftance; and to use the same term to express both it and phlogifton, which, in fact, is no more than the principle of inflammability, must tend to confuse our notions, and betray us into a persuasion, that we know more of the subject than we really do. It will therefore be necessary to keep as near the facts as poffible.

• Phlogiston is that by which bodies, when in contact with pure air, and heated to a certain degree, are put into a ftate of combustion, during which they are in a great measure decompounded, and most commonly, or perhaps universally, exhibit an appearance of fame.

"To vindicate the admission of the cause of inflammability, or phlogiston, among the elements, it will be necessary to shew, that it is a substance, and not a mere modification of the parts of bodies; and that it is so universally similar to itself as to be easily distinguished in all the various bodies which are combu Aible. For this purpose, let us attend to all the circumstances attending combustion, and from thence make our inferences.

• Firtt, All combustible substances, which can be exposed to great heat in close vessels, cannot, in that situation, be calcined or burnt; and in the open air the calcination is more quickly effe&ted accordingly as the supply of pure air is greater. Secondly, If a combustible substance be inflamed, and afterwards included in a vessel, with a small quantity of atmospherical air, the combuftion lafts for a certain time, and then ceases; this time, and consequently that part of the mass which becomes calcined, is greater or less according to the quantity of pure air included; and, al che end of this time, the air is found to be decomposed, part

being fixed and precipitated, and the remainder, which is about four-fifths of the original quantity, is so changed, as to be unfit for the purpose of ailifting combustión, and is in some degree noxious. Thirdly, Sulphur may by conibustion be deprived of its inflammable principle, by which means it is converted into the vitriolic acid ; and again, the vitriolic acid, being properly treated with any inflammable substance, may regain the phlogiston, and be converted into fulphur, which sulphur is poffelled of the fame properties, however different the properties of the inflammable substances may have been by which the phlogiston was furnished. Lastly, The calces of fixed compounds, as metals, may be restored to their original Itate, by being treated with some inflammable substance; and the metal is in all cases the same, however various the phlogistic substances may have been by which it was revived.

· The natural deduction from these facts appears to be, that phlogiston is a substance which is very simple and fimilar to itfelf. For what can the enclosing a combustible body in a close vessel do, but prevent the dispersion of some fubstance? If calcination be no more than a change in the arrangement of the parts of bodies, wby should not heat affect [effect] this as well in close as open veffels ? Is it not evident in the second instance, that the phlogiston becomes combined with the decompounded fuid air, and causes it to deposite one of its principles in the form of fixible air, while the calcination of the i: Hamed body goes forward ; and does not the combustion cease when the remaining principle, or principles of the air, being faturated, are incapable of receiving any more phlogifton? When sulphur, by combustion, is transmuted into vapours, which, when condensed by means of water, and afterwards concentrated, are found to be the vitriolic acid, ought we not to conclude, that this very inflammable substance is converted into a subitance which is not at all so, by having lost one of its principles, namely, phlogiston? But when, on the other hand, the converse of the problem is effected, by producing sulphur from a wellmanaged combination of the vitriolic acid with the inflammable principle of some other body, we can hardly entertain a doubt of the existence of this substance, and at the same time learn, from its effects, that it is similar in all infammable bodies. And this fimilarity of the phlogiston is still more evident from the reduction of the calces of metals; for if a metalline calx be properly treated with any inflammable substance, the external ais being excluded, the metal, from which the calx was originally made, will be reproduced.

• The phlogiston of infiammable bodies being separablc only when thole bodies are in contact with others with which it can combine, it is not probable that we fall ever be able to obtain

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