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*s gered by it. But as for me, neither this, “nor any other difficulty, shall have so great « influence on me, as to make me renounce «s that, which I know to be manifestly agree16! able to reason, especially, when, as it here “ falls out, the difficulty is founded on the per s culiar nature of a certain odd and peculiar a cafe. For in the present case something pecu“ liar lies hid, which being involved in the sub« tilty of nature, will perhaps hardly be dif“s covered, till such time as the manner of Vio “ fion is more perfectly made known.
Concerning the very fame case, another wri. ter in his treatise of Dioptrics says, “And so " he (i: e. Dr. Barrow) leaves this difficulty to as the folution of others, which I (after so great “ an example) shall do likewise, but with the " refolution of the same admirable Author, of “ not quitting the true doctrine, which we have « before laid down, for determining the locus « objecti, on account of being presled by one
difficulty, which seems inexplicable, till a $ more intimate knowledge of the visive fa« culty be obtained by mortals. · The same question may be put, and the same inference made from this case, as froin the former, which it is needless to repeat: And you are not unacquainted, that this important difficulty is now entirely removed, and the matter fairly explained by a learned Prelate, well known to the world for many ingenious performances, as well as his new theory of vision; which should give us hopes, that some difficulties now. belonging to the sciences may hereafter be explained.
ARCHIMEDES is spoken of as one of the greatest Geometricians of the antients, and very justly ; yet let any one examine the demonstrations, which the editors of his works give us, under the title of fele&t theorems of Archimedes, and they will not be found to be accurately true. For as axioms they say : The ambitus of a polygon inscribed in a circle, is less than the periphery of the circle.
And the ambitus of a polygon circumscribed about a circle, is greater than the periphery of the circle,
Yet the third proposition contradicts these, which is, the ambitus of Polygons circumfcribed about and inscribed in a circle, terminate 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 say, they are always greater or less. · The modern Archiinedes (Sir Isaac NEWTON) has used the fame method of demonstration, and proceeds exactly on the same principles. For the first Lemma of his book of mathematical principles of natural philosophy is : Quantities, and the ratio of quantities, which in any finite time converge continually to equality, and before the end of that time, approachnearer the one to the other, than by any given difference, become ultimately equal. This is plainly the same, tho' in other words, with the principles of the antient Archimedes; and the lemma following, concerning inscribed and circumscribed Parallelograms, and a curvilineal
figure coming to a ratio of equality, is a proof of it. .
If you deny the truth of these propositions, 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 consideration of things, and you acquire a most noble fund of knowledge, which has deservedly raised the glory of those men so much above others, as almost to deserve the Poet's praise, Viros supereminet omnes.
But since the Geometricians have made so great a part of this address, it may not be improper also to take notice of the doctrine of Fluxions, which is reckoned the acme of mathematics; insomuch that one writer upon that part of science asks with a kind of exstacy, Whether the next step will not be to the algebra of pure intelligence? He having before produced a testimony, that all the improvements of natural philosophy 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 question, whether the doctrine of Fluxions be scientific or not? The eminent writer, who undertook to prove, that it was not scientific, had both excellent abilities to execute, and an honourable intention in what he did. And the use, that should be made of the dispute, which arose upon that head, is ; that although the doctrine of Fluxions should not be scientific, yet it is a reasonable way of coming at knowledge, and the
mathematicians are wife, in ftill adhering to an useful method of improving natural philofoz phy by Fluxions, as well as in not rejecting Euclid upon account of the extraordinary difficulties attending the proposition mentioned, It is prefumed, no more was intended by examining strictly into the foundation of Fluxions, than to Thew the great necessity for modern ty in all sorts of reasoning concerning revealed propofitions, when the defenders even of Fluxions stand in need of it. For unless the student will out of good nature allow his teacher, to dismiss some embarrassing quantities as of no value, which really have one, he can not make, what he calls a demonstration. It is hoped therefore that they, who are fu conversant in difficulties, which attend the sciences, where the success of human wit has given occafion to human pride, will hereafter judge modestly, both of what, they think, they know, of human science, and what they ought to know of the divine ; and be as little averse to affent to my, Iteries upon the word of God, as the word of man. Éspecially since in molt cases of nicety, human' reason being puzzled, the human mind chuses to rest 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 whose knowledge the writer of these discourfes lias a very high eftcem ; ąs may appear from many parts of these peşformances, which he never could have been able to have executed, without some acquaintance with their mysteries ; inasmuch as
natural philosophy, as now studied and greatly improved, is closely connected with mathém matics: And natural philosophy improved may justly be called, the first chapter of the book of Theology; which divines should highly esteem, as long as there are pretended atheists and infidels in the world : yet with what humility and reasonable allowance for defect, demonstrations in this application are to be received, may be learned from the caution of one, who stands in the first rank of those, who have applied mathematics to the laws of matter. « I would not have any one in phy“ fical matters infift so much on a rigid meof thod of demonstration, as to expect the prin“ ciples of demonstrations, that is axioms, « so 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 such. But we think it sufficient, “ if we deliver such, as we apprehend are « congruous to reason and experience, whose “ truth shines out as it were at first view, which «s procure the belief of such as are not obsti“ nate, to which no body can deny affent, un« less he professes himself to be altogether a es sceptick. But also in demonstrating, it is « necessary to make use of a more lax fort of " reasoning, and to exhibit propofitions that « are not absolutely true, but nearly approach “ ing to truth. As for example, when it is
of demonstrated that all vibrations of the same .« pendulum made in the small arches of a "? circle are of equal duration ; it is here sup