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.]]
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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 15, 2024.

Pages

¶An other example.

This example haue I set to declare

[illustration]
that although the proportion of the first to the second be greater then the proportion of the third to the fourth, yet the multiplex of the first excedeth not the multiplex of the secōd. Wher∣fore it is diligently to be noted,* 1.1 that it is sufficient to shew that the proporti∣on of the first to the second is greater thē the proportion of the third to the fourth, if the want or lacke of the multiplex of the first from the multiplex of the second, be lesse then the want or lacke of the multiplex of the third to the multiplex of the fourth. As in this example 16. the multiplex of 8. the first, wanteth of 20. the multiplex of 4. the second, foure: wheras 18. the multiplex of 9, the third, wāteth of 45, the multiplex of 9 the fourth, 27. And so of all others wheras (the proportions being diuers) the equimultiplices of the first and the third are both lesse, then the equimultiplices of the second and the fourth. Likewise if the equimultiplices of the first and the third do both excede the equimultiplices of the second & the first, thē shall the excesse of the multiplex of the first aboue the multiplex of the second, be greater thē the excesse of the multiplex of the third, aboue the multiplex of the fourth. As in these numbers here set, the equimultiplices of 6. and 4. the first and the third, namely, 12. and 8. do both excede the equimultiplices of 2. and 3. the se∣cond and the fourth, namely, 4. and 6. But 12. the multiplex of the first excedeth 4. the multiplex of the second by 4, and 8. the multiplex of the thyrd excedeth 6. the multiplex of the fourth by 2. but 8. is
[illustration]
more then 2. Howbeit this is general∣ly certaine that when soeuer the pro∣portion of the first to the secōd is grea∣ter then the proportion of the third to the fourth, there may be found some multiplication, that whē the equimul∣tiplices of the first and the third shall be compared to the equimultiplices of the second and the fourth, the multiplex of the first shall excede the multiplex of the second, & the multiplex of the third shall not excede the multiplex of the fourth, according to the plaine wordes of the de∣finition.

In like maner when you haue taken the equimultiplices of the first & the third, and also the equimultiplices of the second and the fourth, if the multiplex of the first excede not the multiplex of the second, and the multiplex of the third excede the multiplex of the fourth: then hath the first to the second a lesse proportion, then hath the third to the fourth. As in the example before, if ye chaunge the termes, and make C the first, D the second, A the third, and B the fourth: then shall

Page 132

F, namely, 12. the multiplex of the first not excede H, namely, 12. the multiplex of the second: but E, namely, 18. the multiplex of the third excedeth G, namely, 8. the multiplex of the fourth. Wherefore the proportion of C to D, the first to the se∣cond, is lesse then the proportion of A to B, the third to the fourth.

Euen so in numbers. As in this ex∣ample,

[illustration]
5. to 4. and 7. to 3. If ye multiply 5. and 7. the first and the third eche by 3, ye shall for the multiplex of 5. the first haue 15. and for the multiplex of 7. the third shall ye haue 21: againe if ye multiply 4. and 3. the second & the fourth by 6, for the multiplex of 4. the second ye shall haue 24, and for the multiplex of 3. the fourth, ye shall haue 18. So ye see that 5. the multiplex of the first, is lesse then 24, the multiplex of the second: and 21. the multiplex of the third is greater then 18. the multiplex of the fourth. Wherefore the proportion of 5. to 4. the first to the second is lesse then the propor∣tion of 7. to 3. the third to the fourth.

Notes

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