The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed

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Title: The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed
Author: Euclid.
Publication: Imprinted at London :: By Iohn Daye,; [1570 (3 Feb.]]
Rights/Permissions: To the extent possible under law, the Text Creation Partnership has waived all copyright and related or neighboring rights to this keyboarded and encoded edition of the work described above, according to the terms of the CC0 1.0 Public Domain Dedication (http://creativecommons.org/publicdomain/zero/1.0/). This waiver does not extend to any page images or other supplementary files associated with this work, which may be protected by copyright or other license restrictions. Please go to http://www.textcreationpartnership.org/ for more information.
Subject terms: Geometry -- Early works to 1800.
Link to this Item: http://name.umdl.umich.edu/A00429.0001.001
Cite this Item: "The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed." In the digital collection Early English Books Online. https://name.umdl.umich.edu/A00429.0001.001. University of Michigan Library Digital Collections. Accessed June 14, 2024.

Pages

A Corollary.

Hereby it is manifest that the diameter of the sphere, is in power quintuple to the line which is drawen from the centre of the circle to the circumference, on which the Icosahedron is described. And that the diameter of the sphere: is com∣posed of the side of an hexagon, and of two sides of a decagon described in one and the selfe same circle.

Flussas proueth the Icosahedron described, to be cōtayned in a sphere, by drawing right lines from the poynt, a, to the poyntes P and G after this maner.

Forasmuch as the lines ZW, WP are put equal to the line drawen from the centre to the circum∣ference•• and the line drawen from the centre to the circumference is double to the line aW, by con∣struction: therefore the line WP is also double to the same line aW. Wherefore the square of the line WP is quadruple to the square of the line aW (by the corollary of the 20. of the sixth). And those lines PW and Wa contayning a right angle PWa (as hath before bene proued) are subtended of the right line aP. Wherefore (by the 47. of the first) the line aP contayneth in power the lines PW, and Wa. Wherefore the right line aP is in power quintuple to the line Wa. Wherefore the right lines aP, and aQ being quintuple to one and the same line Wa, are (by the 9. of the flueth) equall. In like sorte also may we proue that vnto those lines aP and aQ, are equall the rest of the lines drawen from the poynt a to the rest of the angles R, S, T, V. For they subtend right angles contayned of the line Wa, and of the lines drawen from the centre to the circumference. And forasmuch as vnto the line Wa is equall the line Va, which is likewise erected perpendicularly vnto the other plaine superficies OLMNX: there∣fore lines drawen from the point a to the angles O, L, M, N, X, and subtending right angles at the point Z contayned vnder lines drawen frō the centre to the circumference, and vnder the line aZ, are equal not onely the one to the other, but also to the lines, drawen frō the sayde poynt a, to the former angles at the poynts, P, R, S, T, V••. For the lines drawen frō the centre to the circumference of ech circle are equall, & the line aW is equall to the line aZ. But the line aP is proued equal to the line aQ, which is the halfe of the whole QY. Therefore the residue aY is equall to the foresayd lines aP, aQ &c. Wherefore making the centre the poynt a, and the space one of those lines aQ, aP, &c. extende the su∣perficies of a sphere, & it shal touch the 12. angles of the Icosahedron, which are at the pointes O, L, M. N, X, P, K, S, T, V, Q, Y: which sphere is described, if vpō the diameter QY, be drawen a semicircle, and the sayd semicircle be moued about, till it returne vnto the same place from whense it began first to be moued.