By George Boole
This Elibron Classics e-book is a facsimile reprint of a 1877 version by means of Macmillan and Co., London.
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Additional info for A treatise on differential equations
For, while we can prove that we cannot make any inference as to the comparative length of AE, AF. To judge by Heron’s corresponding note to III. 8, he would, to prove this case, practically prove III. e. prove that, if EA be produced to K and FA to L, from which he would infer that, since AK> AL by the first case, An excellent moral can, I think, be drawn from the note of Heron. Having the appearance of supplementing, or giving an alternative for, Euclid’s proposition, it cannot be said to do more than confuse the subject.
A wider definition of a sector than that given by Euclid is found in a Greek scholiast (Heiberg’s Euclid, Vol. V. p. 260) and in an-Nairīzī (ed. Curtze, p. 112). “There are two varieties of sectors; the one kind have the angular vertices at the centres, the other at the circumferences. ” The exact agreement between the scholiast and an-Nairīzī suggests that Heron was the authority for this explanation. The sector-like figure bounded by an arc of a circle and two lines drawn from its extremities to meet at any point actually appears in Euclid’s book On divisions (πρì διαιρσων) discovered in an Arabic MS.
For let the two circles ABC, ADE touch one another internally at the point A, and let the centre F of the circle ABC, and the centre G of ADE, be taken; I say that the straight line joined from G to F and produced will fall on A. For suppose it does not, but, if possible, let it fall as FGH, and let AF, AG be joined. Then, since AG, GF are greater than FA, that is, than FH, let FG be subtracted from each; therefore the remainder AG is greater than the remainder GH. But AG is equal to GD; therefore GD is also greater than GH, the less than the greater: which is impossible.