CONSIDERATIONS UPON THE ANSWER OF DOCTOR WALLIS TO THE THREE PAPERS OF MR. HOBBES.

Dr. Wallis says, all that is affirmed, is but if we SUPPOSE that, this will follow.

But it seemeth to me, that if the supposition be impossible, then that which follows will either be false, or at least undemonstrated.

First, this proposition being founded upon his Arithmetica Infinitorum, if there he affirm an absolute infiniteness, he must here also be understood to affirm the same. But in his thirty-ninth proposition he saith thus: “Seeing that the number of terms increasing, the excess above sub-quadruple is perpetually diminished, so at last it becomes less than any proportion that can be assigned; if it proceed in infinitum it must utterly vanish. And therefore if there be propounded an infinite row of quantities in triplicate proportion of quantities arithmetically proportioned (that is, according to the row of cubical numbers) beginning from a point or 0; that row shall be to a row of as many, equal to the greater, as 1 to 4.” It is therefore manifest that he affirms, that in an infinite row of quantities the last is given; and he knows well enough that this is but a shift.

Secondly, he says, that usually in Euclid, and all after him, by infinite is meant but, more than 446any assignable finite, or the greatest possible. I am content it be so interpreted. But then from thence he must demonstrate those his conclusions, which he hath not yet done. And when he shall have done it, not only the conclusions, but also the demonstration, will be the same with mine in Cap. XIV. Art. 2, 3, &c. of my book De Corpore. And so he steals what he once condemned. A fine quality.

Thirdly, he says, (by Euclid’s tenth proposition, but he tells not of what book), that a line may be bisected, and the halves of it may again be bisected, and so onwards infinitely; and that upon such supposed section infinitely continued, the parts must be supposed infinitely many.

I deny that; for Euclid, if he says a line may be divisible into parts perpetually divisible, he means that all the divisions, and all the parts arising from those divisions, are perpetually finite in number.

Fourthly, he says, that there may be supposed a row of quantities infinitely many, and continually increasing, whereof the last is given.

It is true, a man may say, (if that be supposing) that white is black: but, if supposing be thinking, he cannot suppose an infinite row of quantities whereof the last is given. And if he say it, he can demonstrate nothing from it.

Fifthly, he says (for one absurdity begets another) that a superficies or solid may be supposed so constituted as to be infinitely long, but finitely great, (the breadth continually decreasing in greater proportion than the length increaseth), and so as to have no centre of gravity. Such is 447Toricellio’s Solidum Hyperbolicum acutum, and others innumerable, discovered by Dr. Wallis, Monsieur Fermat, and others. But, to determine this, requires more of geometry and logic, (whatsoever it do of the Latin tongue), than Mr. Hobbes is master of.

I do not remember this of Toricellio, and I doubt Dr. Wallis does him wrong and Monsieur Fermat too. For, to understand this for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.

In the next place, he puts to me a question as absurd as his answers are to mine. Let him ask himself, saith he, if he be still of opinion, that there is no argument in natural philosophy to prove that the world had a beginning. First, whether, in case it had no beginning, there must not have passed an infinite number of years before Mr. Hobbes was born. Secondly, whether, at this time, there have not passed more, that is, more than that infinite number. Thirdly, 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. Fourthly, whether, if this be an absurdity, we have not then, (contrary to what Mr. Hobbes would persuade us), an argument in nature to prove the world had a beginning.

To this I answer, not willingly, but in service to the truth, that, by the same argument, he might as well prove that God had a beginning. Thus, in case he had not, there must have passed an infinite length of time before Mr. Hobbes was born; but there hath passed at this day more than that infinite 448length, by eighty-four years. And this day, which is the last, is given. If this be an absurdity, have we not then an argument in nature to prove that God had a beginning? Thus it is when men entangle themselves in a dispute of that which they cannot comprehend. But, perhaps, he looks for a solution of his argument to prove that there is somewhat greater than infinite; which I shall do so far as to show it is not concluding. If from this day backwards to eternity be more than infinite, and from Mr. Hobbes his birth backwards to the same eternity be infinite, then take away from this day backwards to the time of Adam, which is more than from this day to Mr. Hobbes his birth, then that which remains backwards must be less than infinite. All this arguing of infinites is but the ambition of school-boys.

TO THE LATTER PART OF THE FIRST PAPER.

There is no doubt if we give what proportion we will of the radius to the arc, but that the arc upon that arc will have the same proportion. But that is nothing to my demonstration. He knows it, and wrongs the Royal Society in presuming they cannot find the impertinence of it.

My proof is this: that if the arc on T V, and the arc R S, and the straight line C D, be not equal, then the arc on T V, the arc on R S, and the arc on C A, cannot be proportional; which is manifest by supposing in D C a less than the said D C, but equal to R S, and another straight line, less than R S, equal to the arc on T V; and anybody may examine it by himself.

I have been asked by some that think themselves 449logicians, why I proceeded upon ⅖ rather than any other part of the radius. The reason I had for it was, that, long ago, some Arabians had determined, that a straight line, whose square is equal to 10 squares of half the radius, is equal to a quarter of the perimeter; but their demonstrations are lost. From that equality it follows, that the third proportional to the quadrant and radius, must be a mean proportional between the radius and ⅖ of the same. But, my answer to the logicians was, that, though I took any part of the radius to proceed on, and lighted on the truth by chance, the truth itself would appear by the absurdity arising from the denial of it. And this is it that Aristotle means, where he distinguishes between a direct demonstration and a demonstration leading to an absurdity. Hence it appears that Dr. Wallis’s objections to my Rosetum are invalid as built upon roots.

TO THE SECOND PAPER.

First, he says that it concerns him no more than other men, which is true. I meant it against the whole herd of them who apply their algebra to geometry. Secondly, he says that a bare number cannot be the side of a square figure. I would know what he means by a bare number. Ten lines may be the side of a square figure. Is there any number so bare, as by it we are not to conceive or consider anything numbered? Or, by 10 nothings understands he bare 10? He struggles in vain, his conscience puzzles him. Thirdly, he says 10 squares is the root of 100 square squares. To 450which I answer, first, that there is no such figure as a square square. Secondly, that it follows hence, that a root is a superficies, for such is 10 squares. Lastly, he says that, neither the number 10, nor 10 soldiers, is the root of 100 soldiers; because 100 soldiers is not the product of 10 soldiers into 10 soldiers. This last I grant, because nothing but numbers can be multiplied into one another. A soldier cannot be multiplied by a soldier. But no more can a square figure by a square figure, though a square number may. Again, if a captain will place his 100 men in a square form, must he not take the root of 100 to make a rank or file? And are not those 10 men?

TO THE THIRD PAPER.

He objects nothing here, but that the side of a square is not a superficies, but a line, and that a square root (speaking of quantity) is not a line, but a rectangle, is a contradiction. The reader is to judge of that.

To his scoffings I say no more, but that they may be retorted in the same words, and are therefore childish.

And now I submit the whole to the Royal Society, with confidence that they will never engage themselves in the maintenance of these unintelligible doctrines of Dr. Wallis, that tend to the suppression of the sciences which they endeavour to advance.

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