But that the arch B I and the radius B C are precisely equal, cannot (how true soever it be) be demonstrated, unless that be first proved which is contained in art. 1, namely, that the strait lines drawn from X through the equal parts of O F (produced to a certain length) cut off so many parts also in the tangent B C severally equal to the several arches cut off; which they do most exactly as far as B C in the tangent, and B I in the arch B E; insomuch that no inequality between the arch B I and the radius B C can be discovered either by the hand or by ratiocination. It is therefore to be further enquired, whether the strait line A V cut the arch of the quadrant in I in the same proportion as the point C divides the strait line B V, which is equal to the arch of the quadrant. But however this be, it has been demonstrated that the strait line B V is equal to the arch B H D. [Sidenote: The third attempt; and some things propounded to be further searched into.] 4. I shall now attempt the same dimension of a circle another way, assuming the two following lemmas. Lemma I. If to the arch of a quadrant, and the radius, there be taken in continual proportion a third line Z; then the arch of the semiquadrant, half the chord of the quadrant, and Z, will also be in continual proportion. For seeing the radius is a mean proportional between the chord of a quadrant and its semichord, and the same radius a mean proportional between the arch of the quadrant and Z, the square of the radius will be equal as well to the rectangle made of the chord and semichord of the quadrant, as to the rectangle made of the arch of the quadrant and Z; and these two rectangles will be equal to one another. Wherefore, as the arch of a quadrant is to its chord, so reciprocally is half the chord of the quadrant to Z. But as the arch of the quadrant is to its chord, so is half the arch of the quadrant to half the chord of the quadrant. Wherefore, as half the arch of the quadrant is to half the chord of the quadrant (or to the sine of 45 degrees), so is half the chord of the quadrant to Z; which was to be proved. Lemma II. The radius, the arch of the semiquadrant, the sine of 45 degrees, and the semiradius, are proportional. For seeing the sine of 45 degrees is a mean proportional between the radius and the semiradius; and the same sine of 45 degrees is also a mean proportional (by the precedent lemma) between the arch of 45 degrees and Z; the square of the sine of 45 degrees will be equal as well to the rectangle made of the radius and semiradius, as to the rectangle made of the arch of 45 degrees and Z. Wherefore, as the radius is to the arch of 45 degrees, so reciprocally is Z to the semiradius; which was to be demonstrated.