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 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.