By Thomas L. Heath
After learning either classics and arithmetic on the collage of Cambridge, Sir Thomas Little Heath (1861-1940) used his time clear of his task as a civil servant to submit many works almost about old arithmetic, either renowned and educational. First released in 1926 because the moment variation of a 1908 unique, this e-book includes the second one quantity of his three-volume English translation of the 13 books of Euclid's parts, protecting Books 3 to 9. This exact textual content might be of worth to someone with an curiosity in Greek geometry and the historical past of arithmetic.
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Extra info for Thirteen Books of Euclid's Elements. Books III-IX
8. Euclid's proof contains an unproved assumption, namely that the lines bisecting BG, BH at right angles will meet in a point P. For a discussion of this assumption see note on IV. 5. PROPOSITION I I. If two circles touch one another intenzally, and their centres be taken, the straight line joini·ng their centres, if it be also produced, will fallon the point of contact of the circles. 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 H the centre G of ADE, be taken; I say that the straight line joined from G to F and produced will fall anA.
Therefore the straight line joined from F to G will not fall outside; therefore it will fall at A on the point of contact. Therefore etc. Q. E. D. 2. evyvup-EvT} eMeta). 3. point of contact is here ,;uvarpf}, and in the enunciation of the next proposition brarpf}. Again August and Heiberg give in an Appendix the additional or alternative proof, which however shows little or no variation from the genuine proof and can therefore well be dispensed with. The genuine proof is beset with difficulties in consequence of what it tacitly assumes in the figure, on the ground, probably, of its being obvious to the eye.
The lesser circle is 'within the other. The proof is that of prop. 2 above, mutatis JIlutandis. The circles here touch tl1ternally at the point on the line of centres. 5. If the distance between the centres of lwo circles is less than the sum, and greater tlmn the difference, of the radiI; the two circutliferences haz/e two common points symmetrically situated with respect to the line of centres but not lying on that line. Let 0, 0' be the centres of the two circles, r, 1" their radii, r' being the greater, so that 1" - r < 00' < r + r'.