An Answer to Four Papers of Mr. Hobs, Lately Published in the Months of August, and This Present September, 1671

An Answer to Four 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. I. 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 according 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 a row of as many, equal to the last, 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 SUPPOSE That; This will Follow. Which Consequence Mr. Hobs doth not deny; and therefore all that he saith to it, is but Cavilling.) Mr. Hobs moves these Questions, (and propogeth them to the Royal Society, to pass a judgment on 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 Proposition is not at all concerned, (which affirms nothing either 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 only say (upon supposition) If he were, what he is not; he would not do as he doth. 2. Many of these Quere'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 supposition of things impossible, to infer useful Truths.) And Euclide (in his second Postulate) requiring, the producing a straight line Infinitely, either way; did not mean, that it should be actually performed. (For it is no: possible for any man to produce a straight 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, 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 whoever doth thus suppose Infinites; 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, Infinitely: it is not his meaning (when such continual section is proposed) that it should be actually done, (for, who can do it?) but that it be supposed. And upon such (supposed) section infinitely continued, the parts must be (supposed) infinitely many; for no Finite number of parts would suffice for Infinite 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 mc, Mc, &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 (BC) is given: (and their Squares, as 1, 4, 9, &c. their Cubes, as 1, 8, 27, &c.) And this I say, to shew that the supposition of Infinites (with these attendants) is not so new, or so Peculiar to Cavallerius or Dr. Wallis, but that Euclide admits it, and all Mathematicians with him; as at least supposable, whether Possible or not. In particular, therefore, to his Quere's, I answer, 1. There may be supposed a row of Quantities Infinitely many, and continually increasing, (as the supposed parallels in the Triangle ABC, reckoning downwards from A to BC,) whereof the last (BC) is given. 2. A Finite Quantity (as AB) may be supposed (by (by such continual Bisections) divisible into a number of parts Ininitely many (or, more than any Finite number assignable:) For there is no stint beyond which such division 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 greater than another: As a, sup- posed, infinite number of Men, may be supposed to have a Greater number of eyes. 4. A surface, or solid, may be supposed so constituted, as to be Ininitely Long, but Finitely Great, (the Breadth continually Decreasing 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 innu- merable, discovered by Dr. Wallis, Monsieur Fermat, and others. But to determine this, requires more of Geometry and Logick than Mr. Hobbs is Master of. 5. There may be supposed a number In- finite; that is, greater than any assignable Finite: As the sup- posed number of parts, arising from a supposed Section Ininitely continued. 6. There is therefore no reason, on this account, why the Doctrin of Euclid, Cavalierius, or Dr. Wallis, should be re- jected as of no use. But having solved these Quere's, I have some for Mr. Hobbs to answer, which will not so easily be dispatched by him. For though Supposed Infinites will serve the Mathematicians well e- nough: yet, howsoever he please to prevaricate (which, he faith, is for his Exercise,) Mr. Hobbs himself is more concerned than they, to solve such Quere'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. Hobbs 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 hi- thereto, the last is given. 4. Whether, if this be an Absurdity, we have not then (contrary to what Mr. Hobbs would persuade us) an Argument in nature to prove, the world had a beginning. Nor are we beholden to Mr. Hobbs for this Argument; for it was an Ar- gument in use before Mr. Hobbs was born. Nor can he serve him- self (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 Quare's of Infinites. And this I say, to shew that Mr. Hobs is, as much as any, concerned to solve the Quare's by himself proposed. 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 \( \frac{1}{2} DC \), 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 Confutation of his Rosetum, published in the Philosophical Transactions for July last past. As it is now in the English; his Demonstration is peccant 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 continual 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 \( \frac{DC}{CA} \) are in the same continual proportion, DR, RS, Arc on RS extended \( \frac{DR}{RS} \) portion, of the Semidiameter DT, TV, Arc on TV extended \( \frac{TV}{DT} \) to the Quadrantal Arc; Let that proportion be what you will; suppose, as * See Tab. I. I 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, DT, DR, DC, will be in the same proportion to one another, as (their singles) DT, DR, DC: But these (by construction) 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 portion (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, in any continual proportion (greater, less, or equal to his) the Arcs on tu, on rs, on CA, (extended) will be in the same continual proportion. But (which is the common fault of Mr. Hobbs's Demonstration) if this Demonstration were good, it would serve as well for any proposition as that for which he brings it. For if, instead of \( \frac{1}{2} \), he had said, \( \frac{3}{4}, \frac{1}{2}, \frac{1}{100} \), or what else he pleased; the Demonstration had been just as good as now it is, without changing one syllable: That is, it will equally prove the proportion of the Semidiameter to the Quadrantal Arc, to be, what you please: As any may presently see, who doth but read over his Paper. In his second Paper, He pretends to confute a Theorem, which hath a long time passed for truth; (and therefore doth no more concern Dr. Wallis than other men.) And 'tis this, The four sides of a square being divided into any number of equal parts, for example, into 100; and straight lines drawn through the opposite points, which will divide the Squares into 100 lesser Squares: The received opinion (faith he) and which Dr. Wallis commonly useth, 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 therefore the Root of 100 Squares is 10 of those Squares, and not the side of any Square; because the side of a Square is not a Superficies, but a Line. For An./ 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 Hobbs 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 Hobbs 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 Douzen is the Root, not of 100 Douzen, but of 100 Douzen douzen, or Squares of a Douzen. And, as, there, there, you must multiply not only 10 into 10, but Douzen into Dou- zen, to have the Square of 10 Douzen; so here 10 into 10 (which makes a 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 product of 10 Soldiers into 10 Soldiers, but of 10 Soldiers into the Number 10; And therefore neither 10, nor 10 Soldiers, 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.* 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 contrary, (Prop. 1.) A Square root, (speaking of Quantity) is not a Line, but a Rect- angle. Other faults, falsities, and contradictions, there are a great many. As for Instance: He tells us first, In the natural Row of Num- bers, as 1, 2, 3, 4, 5, 6, &c. every one is the Square of some number in the same Row; (that is, of some Integer number; which is notoriously false.) This he contradicts in the very next words, But Square numbers (beginning at 1) intermit first two numbers, then four, then six, &c.; so that none of the intermitted numbers is a Square number, nor hath any Square root. (If these intermitted numbers, between 1, 4, 9, 16, &c., be not Squares how is it that every one in the whole row is a Square, and that of some Integer number?) But this again is contradicted prop. 2. where 200 (one of such intermitted numbers) is made a Square, and 14\(\frac{4}{14}\) the Root of it. Again; in his Definition he tells us, that a Square Root multiplied into itself produceth a Square: But (prop. 2.) he multiplieth the Root 14\(\frac{4}{14}\) (not into itself, but) into 14 (a part thereof,) to make 200, which he will have to be the Square of that Root. Nor is it a meer slip of negligence in the computation, but his Rule directs to it; Any number given is produced by the greatest Root multiplied into itself, and into the remaining Fraction. Whereof he gives this instance: Let the number given be 200 Squares, the greatest Root is 14\(\frac{4}{14}\) Squares (he should rather have said Lengths; but that is a small fault with him;) I say, that 200 is equal to the product of 14 into itself (which is 196,) together with 14 multiplied into \(\frac{4}{14}\) (which is equal to 4:) that is 14\(\frac{4}{14}\) multiplied into 14. But this calculation is again contradicted in his third proposition, where he calculates the same Square otherwise, as we shall see by and by. In the mean time let's consider this alone, and see the contradictions within it self. His Rule bids us multiply the greatest Root into itself, &c. This greatest Root he says is 14\(\frac{4}{14}\); yet doth he not multiply this, but 14 (a part thereof) into itself and into the Fraction \(\frac{4}{14}\). Again; if 14\(\frac{4}{14}\) be the greatest Root, what shall be the remaining Fraction? Doth he take the Root of 200 to be more than 14\(\frac{4}{14}\) by some further remaining Fraction? If so, he should have told us what that Fraction is; for \(\frac{4}{14}\) it is not, this being part of his greatest Root 14\(\frac{4}{14}\). But if we should allow (as I think we must,) that by the greatest Root he means sometimes 14\(\frac{4}{14}\), sometimes 14\(\frac{2}{14}\) (that is, if we allow him to contradict himself,) yet how comes he by the Fraction \(\frac{4}{14}\)? For, \(\frac{2}{14}\) is too much (the square of 14\(\frac{2}{14}\) being more than 200, as by multiplying 14\(\frac{2}{14}\) into itself will appear;) which destroys his whole design; for 14\(\frac{2}{14}\) multiplied into 14\(\frac{2}{14}\), will not make 200, but 198; contrary to his rule. But further, it is so grots a mistake, to make 200 the Square of 14\(\frac{2}{14}\), that every Apprentice boy, (that can but multiply whole numbers, and fractions,) could have informed him better, who would first have reduced the fraction to smaller terms, putting $\frac{14}{7}$ for $\frac{14}{4}$, and then multiplying $\frac{14}{7}$ into itself, would have shew'd him, that the square of $\frac{14}{7}$, that is, $\frac{14}{7}$ multiplied into itself, is (not 200, but) $204\frac{4}{9}$. But the Root of 200, is the said number $10\sqrt{2}$, which is less than $\frac{14}{7}$, and bigger than $\frac{14}{5}$: the Square of that being somewhat more than 200; and, of this, somewhat less; but either of them within an unite of it. But this second Proposition, is (as I said) contradicted by his third, which makes the Square of $\frac{14}{7}$ to be $200\frac{4}{9}$ (by what computation, we shall see by and by;) and then finds fault, that this and the former do not agree. (But 'tis no wonder they should disagree, when both are false.) The same Square (faith he) calculated Geometrically, consisteth (by Euclid.2.4.) of the same numeral great Square 196, and of two Rectangles under the greatest side 14 and the Remainder of the side, and further of the Square of the less segment; which altogether make $200\frac{4}{9}$. (He might have learned to reckon better; but let us see how he makes it out.) As by the operation itself (faith he) appeareth thus: The side of the greater segment is $\frac{14}{7}$ (this was, but now, the side of the whole square, how comes it now to be but the side of the greater Segment?) which multiplied unto itself (faith he) makes 200: (no; but $204\frac{4}{9}$:) The product of 14 the greatest Segment into the two Fractions $\frac{1}{4}$ is 4, and that added to 196 makes 200: (if by two fractions $\frac{1}{4}$, he mean, as he ought by his Rule, the Fraction 4 twice taken, or the double of it, it will be not 4, but 8, and this added to 196 make 204; But all this he puts in his pocket, for it comes not into account at all.) Lastly, the product of $\frac{1}{4}$ into $\frac{1}{4}$, or $\frac{1}{7}$ into $\frac{1}{7}$ is $\frac{1}{9}$; which with the first 200 makes $200\frac{4}{9}$: (But he forgets himself, for his lesser segment was not $\frac{1}{4}$, but $\frac{1}{4}$; he should therefore have said $\frac{1}{4}$ into $\frac{1}{4}$, or $\frac{1}{7}$ into $\frac{1}{7}$ is $\frac{1}{9}$.) His calculation therefore should have been this: The greater segment is (not $\frac{14}{7}$, but) 14; which multiplied into itself makes (not 200, but 196: The Rectangle of the greater segment 14, into the lesser $\frac{1}{4}$, is 4: And this taken a second time, is another 4: The lesser segment (not $\frac{1}{4}$, but) $\frac{1}{4}$, or $\frac{1}{7}$, multiplied into itself, is (not \( \frac{1}{9} \), but \( \frac{4}{9} \)): All which added together make not 200\( \frac{1}{9} \), but 196 + 4 + 4 + \( \frac{4}{9} \) = 204\( \frac{4}{9} \), which is just the same with 14\( \frac{4}{9} \) multiplied into itself. So that, had he known how to multiply a number into a number, especially when incumbered with fractions (which it is manifest he doth not,) he would have found no disagreement between the Arithmetical calculation, and what he calls the Geometrical. But I am ashamed (for him) that so great a pretender to such high things in Geometry, should be so miserably ignorant of the common operations of practical Arithmetick. His repeated Quadrature he now expresseth thus, The Radius of a Circle is a mean Proportional between the Arc of a Quadrant and two fifths of the same. But instead of two fifths, he might as well have said the half, or tenth, or hundredth part, &c.; or (taking T in DC produced beyond C,) the double, decuple, centuple, &c., or what you please: For his Demonstration would have proved it, which is this. Describe a Square ABCD, and in it a Quadrant DCA. In the side DC (continued if need be,) take DT two fifths of DC, (or its Half, Double, Hundredth part, or what you please;) and between DC and DT a mean proportional DR; and describe the Quadrantal Arcs RS, TV. I say, the Arc RS is equal to the straight line DC. For seeing the proportion of DC to DT is duplicate of the proportion of DC to DR, it will be also duplicate of the Proportion of the Arc CA to the Arc RS, and likewise duplicate of the Proportion of the Arc RS to the Arc TV. Suppose some other Arc, less or greater than the Arc RS, to be equal to DC, as for example RS; Then the proportion of the Arc RS to the straight line DT will be duplicate of the proportion of RS to TV, or DR to DT, which is absurd, because DR is by construction greater or less than DR. Therefore the Arc RS is equal to the side DC; which was to be demonstrated. Which demonstration therefore proving indifferently every proportion, doth not indeed prove any. In brief: The force of his Demonstration is but this; DT being to DC as 2 to 5 (or in any other proportion) and DR a mean proportional between them; RS will be so between TV and CA; and therefore RS (greater or less than RS,) will not be a mean proportional between TV and CA: which is true; but why it may not be equal to DC, we have nothing but his word for it; there being nothing to shew, that DC is equal to such a mean proportional. Again; though RS be not a mean proportional between TV and CA, yet it may be between TV and CA, which serves his Demonstration as well; which is indifferent to any three continual proportionals, as was shewed before. So that now we have had three Demonstrations of this Quadrature, (in his Rosetum, in his first paper, and in his third,) and this common fault in all of them, that they equally prove the proportion by him proposed, or any other what you please. But such his Demonstrations use to be. And this is what I thought fit to say to Mr. Hobs's Four Papers (rather to satisfy the importunity of others, than because I thought them worth Answering:) And submit the whole, with all Respects, to the Royal Society, to whom Mr. Hobs makes his Appeal. His Fourth Paper: Which came out since the Three former were answered, (containing some faint endeavors to re-assert some things in them,) is but meer Trifling, or worse than so. What he would therein insinuate concerning God (that we may as well prove Him to have had a Beginning, as that the World had) smells too rank of Mr. Hobs. We are not to measure Gods Permanent Duration of Eternity, by our successive Duration of Time: Nor, his Intire Ubiquity, by Corporeal Extension. What in it concerns Mathematicks, (whether his own or others,) is so weak and trivial, (and said only, that he may seem to say something, though nothing to the purpose,) that I shall trust it with those to whom he makes his appeal, without thinking it to need any Reply; The view of what he writeth against, being a sufficient Answer to all he saith. New Observations of Spots in the Sun; made at the Royal Academy of Paris, the 11th and 13th of August 1671; and English'd out of the French, as follows. It is now about twenty* years since, that Astronomers have not seen any considerable spots in the Sun, though before that time, since the Invention of Telescopes, they have from time to time observed them. The Sun appeared all that while with an entire brightness, and Signor Cassini saw him to * See NumL.74.p.2216; whence it will appear, that some such spots were seen here in London, A.1660. And Mons. Picard affirmed to Dr. Fogelius at Hamburg, that he had seen some in October 1661, witness the said Doctor's own Letter, written to the Publisher August 1st last.