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

When the equemultiplices being taken,* 1.1 the multiplex of the first excedeth the multiplex of the second, & the multiplex of the third, excedeth not the multiplex of the fourth: then hath the first to the second a greater proportion, then hath the third to the fourth.

In the sixt definition was declared what magnitudes are said to be in ••••e and the same proportion: now he sheweth in this definition what magnitudes are said to be in a greater proportion. And here is supposed the same order of multiplicati∣on, that there in that definition was vsed: namely, that the first and the third be e∣qually multiplied, that is, by one & the selfe same nūber: and also that the second and the fourth be equally multiplied by

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the same or some other number: and then if the multiplex of the first, excede the multiplex of the second:* 1.2 & the mul∣tiplex of the third; excede not the mul∣tiplex of the fourth, the first hath a grea∣ter proportion to the second, then hath the third to the fourth. As suppose that there be foure quantities, A, B, C, D: of which let A be the first, B the second, C the third, & D the fourth. And let A the first cōmine 6. and let B the second con¦taine 2. & C the third 4. & D the fourth 3: Now take the equimultiplices of A and C the first & the third, which let be E and F, so that how multiplex E is to A so multiplex let F be to C: name∣ly for example sake let either of them be triple: so haue you 18. for the multiplex of A, and 12. for the mul∣tiplex of C. Likewise take the equimultiplices of B & D, the second & the fourth, multiplying them also by one and the self same number, as by 4: so haue ye for the multiplex of B the second 8, namely, the line G, and for the multiplex of D the fourth 12, namely, the line H. Now because the line E multiplex to the first, name∣ly, 18, excedeth the line G multiplex to the second, namely, 8: And the line F mul∣tiplex to the third, namely, 12, excedeth not the line H multiplex to the fourth, namely, 12 (for that they are equall) the proportiō of A to B the first to the second, is greater then the proportion of C to D the third to the fourth. So likewise in nū∣bers: take 11. to 2. & 7. to 3. and mul∣tiply

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11. & 7.* 1.3 (the first, and the third) by 2, so shall ye haue 22. for the multi∣plex of the first, and 14. for the multi∣plex of the third: and multiply 2. and 3. the second and the fourth by 6: so shall the multiplex of the second be 12. and the multiplex of the fourth be 18:

Now ye see 22. the multiplex of the first, excedeth 12, the multiplex of the second. But 14. the multiplex of the third, excedeth not 18. the multiplex of the fourth: Wherefore the proportion of 11. to 2. the first to the second, is greater then the pro∣portion of 7. to 3, the third to the fourth. And so of all other quantities and num∣bers, which are not in one and the selfe same proportion, ye may know when the first to the second hath a greater proportion then the third to the fourth.

¶An other example.

This example haue I set to declare

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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,* 1.4 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

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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

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,

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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

• * 1.1

  The eight definition.
• * 1.2

  An example of this defini∣tion in mag∣nitudes.
• * 1.3

  An example in numbers.
• * 1.4

  Note.