An answer to three papers of Mr. Hobs lately published in the months of August, and this present September, 1671.
Wallis, John, 1616-1703.

In the former part of his first Paper;

BY reason of a Proposition of Dr. Wallis (Prop. 1. Cap. 5. De Motu) to this purpose (for he doth not repeat it Verbatim:) If there be supposed a row of Quantities infinitely many, increasing ac∣cording to the natural Order of Numbers, 1, 2, 3, &c. or their Squares, 1, 4, 9, &c. or their Cubes, 1, 8, 27, &c. whereof the last is given. It will be to a row of as many, equal to the lst, in the first case, as 1 to 2; in the second case, as 1 to 3; in the third▪ as 1 to 4, &c. (Where all that is affirmed, is but; If we SVPPOSE That, This will Follow. Which Con∣sequence Mr. Hobs doth not deny: and therefore all that he saith to it, is but Cavelling.)

Mr. Hobs moves these Questions, (and proposeth them to the Royal Society, as not requiring any skil in Geometry, Logick, or Latin, to resolve them:) 1. Whether there can be understood (he should rather have said, supposed) an infinite row of Quantities, whereof the last can be given. 2. Whether a Finite Quantity can be divided into an Infinite Number of lesser Quantities, or a Finite quantity consist of an Infinite number of Parts. 3. Whether there be any Quantity greater than Infinite. 4. Whether there be any Finite Magnitude of which there is no Center of Gravity. 5. Whether there be any Number Infinite. 6. Whether the Arithmetick of Infinites be of any use, for the confirming or confuting any Doctrine.

For answer. In general, I say, 1. Whether those things Be or Be not; yea, whether they Can or Cannot be; the Pro∣position is not at all concerned, (which affirms nothing ei∣ther way;) but, whether they can be supposed, or made the supposition, in a conditional Proposition. As when I say, If Mr. Hobs were a Mathematician, he would argue otherwise: I do not affirm that either he is, or ever was, or will be such. I on∣ly say (upon supposition) If he were, what he is not; he would not do as he doth. 2. Many of these Quaere's have nothing to do with the Proposition: For it hath not one word concerning Gravity, or Center of Gravity, or Greater than Infinite. 3. That usually in Euclide, and all after him, by Infinite is meant but, More than any assignable Finite, though not Absolutely Infinite, or the greatest possible. 4. Nor do they mean, when Infinites are proposed, that they should actually Be, or be possible to be performed; but only, that they be supposed. (It being usual with them, upon sup∣position of things Impossible, to infer useful Truths.) And Euclide (in his second Postulate) requiring, the producing a streight line Infinitely, either way; did not mean, that it should be actually performed, (for it is not possible for any man to produce a streight line Infinitely;) but, that it be supposed. And if AB be supposed so produced, though but one way; its length must be supposed to become Infinite (or more than any Finite length assignable;) For, if but Finite, a Finite production would serve. But,

[illustration]
if so produced both ways; it will be yet Greater, that is, Greater than that Infinite, or Greater than was necessary to make it more than any Finite length assignable. (And who∣ever doth thus suppose Infi∣nites; must consequently suppose, One Infinite greater than another.) Again, when (by Euclide's tenth Proposition) the same AB, may be Bisected in M▪ and each of the halves in m, and so onwards, Infinite∣ly: it is not his meaning when such continual section is pro∣posed) that it should be actually done, (for, who can do it?) but that it be supposed. And upon such (suppsd) section infinitely continued, the parts must be (supposed) infinitly many; for no Finite number of parts would suffice for Infi∣nite sections. And if further, the same AB so divided, be supposed the side of a Triangle ABC; and, from each point of division, supposed lines (as me, Me, &c.) parallel to BC: these parallels (reckoning downward from A to BC) must consequently be (supposed) infinitely many; and those, in Arithmetical progression, as 1, 2, 3, &c. (each exceeding its Antecedent as much as that exceeds the next before it;) and, whereof the last (BCis given. Nor is the supposition of Infinites (with these attendants) so new, or so Peculiar to Ca∣vallerius or Dr. Wallis, but that Euclide admits it, and all Ma∣thematicians with him; as at least supposable, whether Pos∣sible or not.

In particular, therefore, to his Quaere's, I answer, 1. There may be supposed a row of Quantitis Infinitely many, and continually increasing, (as the supposed parallels in the Tri∣angle ABC, reckoning downwards from A to BC,) whereof the last (BC) is given. 2. A Finite Quantity (as AB) may be supposed (by such continual Bisections) divisible into a num∣ber of parts Infinitely many (or, more than any Finite num∣ber assignable:) For there is no stint beyond which such di∣vision may not be supposed to be continued; (for still the last, how small soever, will have two halves;) And, all those Parts were in the Undivided whole; (else, where should they be had?) 3. Of supposed Infinites, one may be supposed grea∣ter than another. As a, supposed, infinite number of Men, may be supposed to have a Greater number of eyes. 4. A sur∣face, or solid, may be supposed so constituted, as to be Infi∣nitely Long, but Finitely Great, (the Breadth continually De∣creasing in greater proportion than the Length Increaseth,) and so as to have no Center of Gravity. Such is Toricellio's Solidum Hyperbolicum acutum; and others innumerable, dis∣covered by Dr. Wallis, Monsieur Format, and others. But to determine this, requires more of Geometry, and Logick (what∣ever it do of the Latin Tongue) than Mr. Hobs is Master of. 5. There may be supposed a number Infinite; that is, grea∣ter than any assignable Finite: As the supposed number of parts, arising from a supposed Section Infinitely continued. 6. There is therefore no reason, on this account, why the Doctrin of Euclide, Cavallerius, or Dr. Wallis, should be re∣jected as of no use.

But having solved these Quaere's, I have some for Mr. Hobs to answer, which will not so easily be dispatched by him. For though Supposed Infinites will serve the Mathematicians well enough: yet, howsoever he please to prevaricate (which, he saith, is for his Exercise,) Mr. Hobs himself is more concerned than they, to solve such Quaere's. Let him ask himself therefore, if he be still of opinion, that there is no Argument in nature to prove, the World had a Beginning: 1. Whether, in case it had not, there must not have passed an Infinite number of years before Mr. Hobs was born. (For, if but Finite, how many soever, it must have begun so many years before.) 2. Whether, now, there have not passed more; that is, more than that infinite number. 3. Whether, in that Infinite (or more than infinite) number of Years, there have not been a Greater number of Days and Hours: and, of which hitherto, the last is given. 4. Whether, if this be an Absurdity, we have not then (contrary to what Mr. Hobs would perswade us) an Argument in nature to prove the world had a beginning: Nor are we beholden to Mr. Hobs for this Argument; for it was an Argument in use before Mr. Hobs was born. Nor can he serve himself (as the Mathematicians do) with supposed Infinites; For his Infinites, and more than Infinites of Years, Days, and Hours, already past, must be Real Infinites, and which have actually existed, and whereof the last is given; (and yet there are more to follow.) Mr. Hobs shall do well (for his Exercise) to solve these, before he propose more Quaere's of Infinites.