An answer to three papers of Mr. Hobs lately published in the months of August, and this present September, 1671.
Wallis, John, 1616-1703.
   
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AN ANSWER TO Three Papers of Mr. Hobs, Lately Published in the Months of August, and this present September, 1671.

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.

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In the latter part of his first Paper,

HE gives us (out of his Roset. Prop. 5.) this Attempt of Squaring the Circle. Suppose DT be DC, and DR a mean proportional between DC and DT: the Semidiameter DC will be equal to the Quadrantal Arc RS, and DR to TV.

That the thing is false, is already shewed in the Latin Con∣futation of his Rosetum, published in the Philosophical Trans∣actions for July last past.

As it is now in the English; his Demonstration is pec∣cant in these words, (Col. 2. lin. 31, 32, 33.) Therefore - the Arc on TV, the Arc on RS, the Arc on CA, cannot be in conti∣nual proportion; (with all that follows:) There being no ground for such Consequence.

[And the thing is manifest; for since that, by his construction,

  • DC.CA. Arc on CA extended ∷ are in the same continual proportion, of the Semidia∣meter to the Quadrantal Arc;
  • DR.RS. Arc on RS extended ∷ are in the same continual proportion, of the Semidia∣meter to the Quadrantal Arc;
  • DT.TV. Arc on TV extended ∷ are in the same continual proportion, of the Semidia∣meter to the Quadrantal Arc;
Let that proportion be what you will; suppose, as 1 to 2; and consequently, DC to CA being as 1 to 2, it will be to the Arc on CA, as 1 to 4: And by the same reason, DR to the Arc on RS, and DT to the Arc on TV, must also be as 1 to 4: And therefore the Arcs on TV, on RS, on CA; that is, 4 DT, 4 DR, 4 DC; will be in the same proportion to one another, as (their singles) DT, DR, DC: But these (by con∣struction) are in continual proportion; therefore those Arcs also, as they ought to be. Indeed, if (by changing some one of the terms) you destroy (contrary to the Hypothesis) the continual proportion of DT, DR, DC, you will destroy that of the Arcs also (which are still proportional to these:) but so long as DT, DR, DC, be in any continual proportion (whether that by him assigned or any other) those will be in the same continual proportion with them. As if for DT, DR, DC, be taken Dt, Dr, DC,
[illustration]
in any continual proportion (grea∣ter, less, or equal to his) the Arcs on tu, on rs, on CA, (extended) will be in the same continual proporti∣on.]

But (which is the common fault of Mr. Hobs's Demonstrati∣on) if this Demonstration were ood, it would serve as well for any proportion as that for which he brings it. For if, instead of , he had said, 〈…〉, or what else he pleased; the De∣monstration had been just as good as now it is, without chan∣ing one syllable: That is, it will equally prove the propor∣ton of the Semidiameter to the Quadrantal Arc, to be, what yu please.

In his second Paper.

HE pretends to confute a Theorem, which hath a long time passed for truth; (and therefore doth no more con∣ern Dr. Wallis, than other men.) And 'tis this, The four sides f a square being divided into any number of equal parts, for ex∣mple, into 100; and streight lines drawn through the opposite oints, which will divide the Square into 100 lesser Squares: The received opinion (saith he) and which Dr. Wallis commonly seth, is, that the Root of those 100, namely 10, is the side of the whole Square. Which to confute, he tells us, The Root 10 is a number of Squares, whereof the whole contains 100; and there∣fore the Root of 100 Squares is 10 of those Squares, and not the sde of any Square; because the side of a Square is not a Super∣cies, but a Line.

For Answer; I say, that 'tis neither the opinion of Doctor Wallis, nor (that I know) of any other (so far is it from being a Received Opinion, which Master Hobs insinuates as such) that 10 is the Root of 100 Squares (For surely a Bare Number cannot be the side of a Square Figure:) Nor yet (as Master Hobs would have it) that 10 Squares is the Root of 100 Squares: But that 10 Lengths is the Root of 100 Squares. 'Tis true that the Number 10 is the Root of the Number 100, but not, of a 100 Squares: and, that 10 Squares is the Root (not of 100 Squares, but) of 100 Squared Squares: Like as 10 Dousen is the Root, not of 100 Dousen, but of 100 Dousen dousen, or Squares of a Dousen. And, as, there, you must multiply not only 10 into 10, but Dousen in∣to Dousen, to have the Square of 10 Dousen; so here, 10 in∣to 10 (which makes 100) and Length into Length (which makes a Square) to obtain the Square of 10 Lengths, which is therefore 100 Squares, and 10 Lengths the Root or side of it. But, says he, the Root of 100 Soldiers, is 10 Soldiers. Answer. No such matter: For 100 Soldiers is not the pro∣duct of 10 Soldiers into 10 Soldiers, but of 10 Soldiers into the Number 10: And therefore neither 10, nor 10 Sol∣diers, the Root of it. So 10 Lengths into the Number 10, makes no Square, but 100 Lengths; but 10 Lengths into 10 Lengths makes (not 100 Lengths, but) 100 Squares.

So in all other proportions: As, if the number of Lengths in the Square side be 2; the number of Squares in the Plain will be twice two, (because there will be two rows of two in a row:) If the number of Lengths in the side, be 3; the number of Squares in the Plain, will be 3 times 3, or the Square of 3: If that be 4, this will be 4 times 4: And so in all other proportions. Of which, if any one doubt he may believe his own eyes.

[illustration]
And this Mr. Hobs might have been taught by the next Car∣penter (that knows but how to measure a Foot of Board) who could have told him, that because the side of a Square Foot, is 12 Inches in Length, the Plain of it will be 12 times 12 Inches in Squares: Because there will be 12 Rows of 12 in a Row.

His third Paper,

WHich came out just as the Answer to the two former was going to the Press, contains, for substance, the same with his Second, and the Latter part of the First: And so needs no farther Answer.

Only I cannot but take notice of his usual trade of contra∣dicting himself. His second Paper says, The side of a Square is not a Superficies, but a Line: His Third says the quite con∣trary, (Prop. 1.) A Square root (speaking of Quantity) is not a Line, but a Rectangle. Other faults, falsities, and contra∣dictions, there are a great many; which I omit, as too gross to need an Answer.

And this is what I thought fit to say to Mr. Hobs's Three Papers (rather to satisfie the importunity of others, than because I thought them worth Answering:) And sub∣mit the whole, with all Respects, to the Royal Society, to whom Mr. Hobs makes his Appeal.