Page:Carroll - Euclid and His Modern Rivals.djvu/179

. § 1.] two secants, in their ultimate positions, will coincide. Hence the phrase 'the tangent' assumes, without proof, Th. 7. Cor. 1, viz. 'there can be only one tangent to a circle at a given point.' This is a 'Petitio Principii.' P. 97. Th. 6. The secant consists of two portions, each terminated at the fixed point. All that you prove here is that the portion which has hitherto cut the circle is ultimately outside: and you jump, without a shadow of proof, to the conclusion that the same thing is true of the other portion! Why should not the second portion begin to cut the circle at the precise moment when the first ceases to do so? This is another 'Petitio Principii.' P. 129, line 3 from end. 'Abstract quantities are the means that we use to express the concrete.' Excluding such physical 'means' as pen and ink or the human voice (to which you do not seem to allude), I presume that the 'means' referred to in this mysterious sentence are 'pure numbers.' At any rate the only instances given are 'seven, five, three.' Now take P. 130, l. 5, 'Abstract quantities and ratios are precisely the same things.' Hence all ratios are numbers. But in the middle of the same page we read that 'all numbers are ratios, but all ratios are not numbers.' I leave this without further remark.

I will now sum up the conclusions I have come to with respect to your Manual.

(1) As to 'straight Lines' you suggest a useful extension of Euclid's Axiom.

(2) As to angles and right angles, your extension of