ARCHIMEDES is fpoken of as one of the greateft Geometricians of the antients, and very justly; yet let any one examine the demonftrations, which the editors of his works give us, under the title of felect theorems of Archimedes, and they will not be found to be accurately true. For as axioms they fay: The ambitus of a polygon infcribed in a circle, is less than the periphery of the circle.

And the ambitus of a polygon circumfcribed about a circle, is greater than the periphery of the circle.

Yet the third propofition contradicts thefe, which is, the ambitus of Polygons circumfcribed about and infcribed in a circle, terminatè in the periphery of the circle. In like manner the polygons themselves end in the circle: that is, become the circle, which is contrary to the axioms which fay, they are always greater or lefs.

The modern Archimedes (Sir ISAAC NEWTON) has used the fame method of demonftration, and proceeds exactly on the fame principles. For the firft Lemma of his book of mathematical principles of natural philofophy is: Quantities, and the ratio of quantities, which in any finite time converge continually to equality, and before the end of that time, approach nearer the one to the other, than by any given difference, become ultimately equal. This is plainly the fame, tho' in other words, with the principles of the antient Archimedes; and the lemma following, concerning infcribed and circumfcribed Parallelograms, and a curvilineal

figure

figure coming to a ratio of equality, is a proof of it.

If you deny the truth of thefe propofitions, and refuse to use them as true, you in a great measure destroy two noble fabrics of science: Use them as true, tho' you know them to be otherwise in the most strict confideration of

things, and you acquire a moft noble fund of knowledge, which has defervedly raised the glory of those men fo much above others, as almost to deserve the Poet's praise, Viros fupereminet omnes.

But fince the Geometricians have made fo great a part of this address, it may not be improper alfo to take notice of the doctrine of Fluxions, which is reckoned the acme of mathematics; infomuch that one writer upon that part of science asks with a kind of exftacy, Whether the next step will not be to the algebra of pure intelligence? He having before produced a teftimony, that all the improvements of natural philofophy of late years, have been almost wholly owing to the doctrine of Fluxions.

Notwithstanding the utility of this part of knowledge to mankind, and the great honour to the inventor, it is now a fair queftion, whether the doctrine of Fluxions be fcientific or not? The eminent writer, who undertook to prove, that it was not fcientific, had both excellent abilities to execute, and an honourable intention in what he did. And the use, that fhould be made of the difpute, which arose on that head, is; that although the doctrine of Fluxions fhould not be fcientific, yet it is a reafonable way of coming at knowledge; and the

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mathematicians are wife, in ftill adhering to an useful method of improving natural philofophy by Fluxions, as well as in not rejecting Euclid upon account of the extraordinary difficulties attending the propofition mentioned, It is prefumed, no more was intended by examining ftrictly into the foundation of Fluxions, than to fhew the great neceffity for modefty in all forts of reasoning concerning revealed propofitions, when the defenders even of Fluxions ftand in need of it. For unless the student will out of good nature allow his teacher, to difmifs fome embarraffing quantities as of no value, which really have one, he can not make, what he calls a demonftration. It is hoped therefore that they, who are fo converfant in difficulties, which attend the sciences, where the fuccefs of human wit has given occafion to human pride, will hereafter judge modeftly, both of what, they think, they know, of human fcience, and what they ought to know of the divine; and be as little averfe to affent to myfteries upon the word of God, as the word of man. Efpecially fince in mott cafes of nicety, human reafon being puzzled, the human mind chufes to reft judgment upon authority; and what authority can come in competition with the divine?

But one word more with the Gentlemen of the scientific profeffion, for whofe knowledge the writer of thefe difcourfes has a very high eftcem; as may appear from many parts of thefe performances, which he never could have been able to have executed, without fome acquaintance with their myfteries; inasmuch as

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natural philofophy, as now ftudied and greatly improved, is clofely connected with mathe matics: And natural philofophy improved may juftly be called, the firft chapter of the book of Theology; which divines fhould highly esteem, as long as there are pretended atheifts and infidels in the world: : yet with' what humility and reasonable allowance for defect, demonftrations in this application are to be received, may be learned from the caution of one, who ftands in the first rank of thofe, who have applied mathematics to the laws of "I would not have any one in phy"fical matters infift so much on a rigid me"thod of demonftration, as to expect the prin

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ciples of demonftrations, that is axioms, "fo clear and evident in themselves, as those "that are delivered in the elements of Geome "try. For the nature of the thing will not "admit of fuch. But we think it fufficient, "if we deliver fuch, as we apprehend are "congruous to reafon and experience, whofe "truth fhines out as it were at firft view, which procure the belief of fuch as are not obstinate, to which no body can deny affent, un"lefs he profeffes himself to be altogether a fceptick. But alfo in demonftrating, it is neceffary to make use of a more lax fort of reafoning, and to exhibit propofitions that "are not abfolutely true, but nearly approaching to truth. As for example, when it is "demonstrated that all vibrations of the fame pendulum made in the fmall arches of a "circle are of equal duration; it is here fup

S pofed

"pofed, that the fmall arch of a circle and its "chord are of the fame declivity, and of the "fame length; which, however if we regard "the rigid truth, is not to be admitted. But "in phyfics, this hypothefis varies fo little from "the truth, that the difference ought juftly ❝to be neglected, and the disagreement of the "vibrations arifing from that difference is al

together infenfible, as is proved by experi"ence. So likewife that eminent philofopher " and Geometer Dr. Gregory, in his elements "of catoptrics and dioptrics makes ufe of a "more lax Geometry, by affuming lines and "angles as equal, that in reality are unequal, "tho' they accede nearly to an equality, and "fo he folves many beautiful phyfical pro"blems, which otherwife would prove very "intricate? And also this method feems to be approved of fometimes, by Sir Ifaac Newton himself, as may be feen in Prop. 3.Lib. 2. of his Phil. Nat. Prin. Math. But if there are any who harden their minds against such principles, and demonftrations, and will not "fuffer themselves to be convinced by propo"fitions fufficiently manifeft; we leave fuch "to their fupine ignorance; nor do we think "them worthy to be admitted to the know«ledge of the true philosophy.

This caution is very prudent, and the final declaration extremely juft, and in the fame manner the preachers of divine knowledge, and almoft in the fame words, may addrefs their hearers.

If there be any who harden their minds against the knowledge of divine matters, which are built

upon