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

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.