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

Definitions.

1. Like rectiline figures are such,* 1.1 whose angles are equall the one to the other, and whose sides about the equall angles are proportionall.

As if ye take any

[illustration]

two rectiline figures. As for example, two triangles ABC, and DEF: 〈…〉〈…〉 of the one triangle be e∣quall to the angles of the other, namely, if the angle A be equall to the angle D, and the angle B equall to the angle E, & also the an∣gle C equall to the an∣gle F. And moreouer, i•• the sides which con∣taine the equall angles be proportionall. As if the side AB haue that proportion to

the side BC, wh••ch the side DE hath to the side EF, and also if the side BC be vnto the side CA, as ••he side EF is to the side FD, and mor••ouer if the side CA be to the side AB, as the side FD is to the side DE, then are these two triangles sayd to be like: and so iudge ye of any other kinde of figures. As if in the paralle∣logrammes ABCD and EFGH, the angle A be equall to the angle E, and the angle B equall to the angle F, and the angle C equall to the angle G, and the an∣gle D equall to the angle H. And farthermore, if the side AC haue that propor∣tion to the side CD which the side EG hath to the side GH, and if also the side CD be to the side DB as the side GH is to the side HF, and moreouer, if the side DB be to the side BA as the side HF is to the side FE, and finally, if the side BA be to the side AC as the side FE is to the side EG, then are these parallelo∣grammes like.

* 1.22. Reciprocall figures are those, when the terme•• of proporti∣on are both antecedentes and consequentes in either figure.

As if ye haue two parallelogrammes

[illustration]

ABCD and EFGH. If the side AB to the side EF, an antecedent of the first figure to a consequent of the second figure, haue mutually the same proportion, which the side EG hath to the side AC an antece∣dent of the second figure to a consequent of the first figure: then are these two figures Reciprocal. They are called of some, fi∣gures of mutuall sides, and that vndoubted∣ly not amisse nor vnaptly.* 1.3 And to make thys definition more plaine, Campane and Pestitarius, and others•• thus put it: Reciprocall figures, are when the sides of other 〈◊〉〈◊〉 mutually proportionall, as in the example and declaration before geuen. Among the barbarous they are called Mutekesia, reseruing still the Ara∣bike worde.

* 1.43. A right line is sayd to be deuided by an extreme and meane proportion, when the whole is to the greater part, as the grea∣ter part is to the lesse.

As if the line AB, be so deuided in the point

[illustration]

C, that the whole line AB haue the same pro∣portion, to the greater part thereof, namely, to AC, which the same greater part AC hath to the lesse part therof, namely, to CB, then is the line AB deuided by an extreme and meane proportion. Commonly it is called a line deuided by proportion ha••ing a meane and two extremes. How to deuide a line in such sort was taught in the 11. Proposition of the second Booke, but not vnder this forme of proportion.

4. The alitude of a figure is a perpendicular line drawen from the toppe to the base.* 1.5

As the altitude or hight of the triangle ABC, is the line AD being drawen perpendicularly from the poynt A, being the toppe or highest part of the triangle to the base therof BC. So likewise in other figures as ye see in the examples here set. That

[illustration]

which here ••ee calleth the alti∣tude or height of a fi∣gure, in the first booke in the 35. Proposi∣tion and certaine other following, he taught to be contayned within two equidi∣stant lines: so that figures to haue one altitude and to be contayned within two e∣quidistant lines, is all one. So in all these examples, if from the highest point of the figure ye draw an equidistant line to the base therof, and then frō that poynt draw a perpendicular to the same base that perpendicular is the altitude of the figure.

5. A Proportion is said to be made of two proportions or more, when the quantities of the proportions multiplied the one into the other, produce an other quantitie.* 1.6

Of addition of proportions, hath bene somewhat sayd in the declaration of the 10. definition of the fift booke: which in substance is all one with that which is here taught by Euclide. By the name of quantities of proportions, he vnderstan∣deth the denominations of proportions. So that to adde two proportions toge∣ther, or more, and to make one of them all, is nothyng els, but to multiply their quantities together, that is to multiply euer the denominator of the one by the denominator of the other.* 1.7 Thys is true in all kindes of proportion, whether it be of equalitie, or of the greater inequalitie, when the greater quantitie is referred to the lesse: or of the lesse inequalitie, when the lesse quantitie is referred to the grea∣ter: or of them mixed together. If the proportions be like, to adde two together is to double the one, to adde 3. like is to triple the one, and so forth in like propor∣tions, as was sufficiently declared in the declaration of the 10. and 11. definitions of the fift Booke. Where it was shewed, that if there be 3. quantities in like pro∣portion, the proportion of the first to the thyrd, is the proportion of the first to the second doubled: and if there be foure quantities in like proportion; the proportion of the f••••st ••o the fourth shall be the proportion of the first to the second ••••ipled•• which thing how to do was there taught likewyse in proportions vnlike, the proportion of the first extreme to the last is made of all the meane proportions set

betwene them. Suppose three quantities A, B, C, so that let A haue to B sesqui∣altera proportion, namely, 6. to 4. And let B to C haue sesquitertia proportion, namely, 4. to 3.* 1.8 Now the proportion of A to

[illustration]

C, the first to the thyrd, is made of the propor∣tion of A to B, and of the proportion of B to to C added together. If ye will adde them to∣gether, ye must by this definition multiply the quantitie or denominator of the one, by the quātitie or denominator of the other. Ye must first therefore seeke the denominators of these proportions, by the rule before geuen in the declaration of the definitions of the fift Booke. As if ye deuide A by B, namely, 6. by 4, so shall ye haue in the quotient 1 ••/•• for the denominator of the proportion of A to B: likewise if ye deuide B by C, namely, 4. by 3. ye shall haue in the quotient 1 ••/•• for the denominator of the proportion of B to C, now multiply these two denominators 1 ••/•• and 1 ••/•• the one into the other, by the rule before taught, namely, by multiplying the numerator of the one into the nu∣merator of the other, and also the denominator of the one into the denominator of the other: the numerator of 1 ••/•• or of ••/•• which is all one, is 3, the denomi∣nator is 2: the numerator of 1 ••/•• which reduced are ••/•• is 4,

[illustration]

the denominator is 3: then multiply 3. by 4, numerator by nu∣merator, so haue ye 12. for a new numerator: likewise multiply 2. by 3. denominator by denominator, ye shall produce 6. for a new denominator: so haue you produced 12. and 6, betwene which there is dupla proportion. Which proportion is also be∣twene A and C, namely, 6. to 3, the first quantitie to the third. Wherfore the pro∣portion of A to C is sayd to be made of the proportion of A to B and of the pro∣portion of B to C, for that it is produced of the multiplication of the quantitie or denominator of the one,* 1.9 into the quantitie or denominator of the other. And so of all others be they neuer so many. As in these examples in numbers here set 2. 3. 15. 18.* 1.10 In this example the lesse numbers are compared to the greater, as in the former the greater were compared to the lesse: the denominator of the proporti∣on of 2. to 3. is 2/•• that is, subsesquialtera, the denomination betwene 3. and 15. is ••/•• or 1/•• which is all one, that is, subquintupla, betwene 15. and 18. the denomi∣nation of the proportion is 1/6 that is, subsesquiquinta, multiply all these denomi∣nations together: first the numerators: 2. into 1. produce 2, then 2. into 5. produce 10: which shall be a new numerator. Then the de∣nominators:

[illustration]

3. into 5. produce 15: and 15. into 6. produce 90: which shall be a new denominator. So haue you brought forth 10/9•• or 1/9 which is pro∣portion subnoncupla: which is also the proportion of 2. to 18. Wherefore the proportion of 2. to 18. that is, of the extremes, namely, subnoncupla, is made of the proportions of 2. to 3: of 3. to 15: and of 15. to 18: namely, of subsesquialtera, subquintupla, and subses∣quiquinta.

An other example, where the greater inequalitie and the lesse inequalitie are mixed together 6. 4. 2. 3. the denomination of the proportion of 6. to 4, is 1 ••/••,* 1.11 of 4. to 2, is ••/••, and of 2. to 3, is ••/••: now if ye multiply as you ought, all these denominations together, ye shall produce 12. to 6, namely, dupla proportion.

Forasmuch as so much hath hetherto bene spoken of addition of proportions

it shall not be vnnecessary somewhat also to say of substraction of them.* 1.12 Where it is to be noted, that as addition of them, is made by multiplicatiō of their denomi∣nations the one into the other: so is the substraction of the one from the other done, by diuision of the denomination of the one by the denomination of the o∣ther. As if ye will from sextupla proportion subtrahe dupla proportion, take the denominations of them both. The denomination of sextupla proportion, is 6, the denomination of dupla proportion, is 2. Now deuide 6. the denomination of the one by 2. the denomination of the other: the quotient shall be 3: which is the denomination of a new proportion, namely, tripla: so that when dupla pro∣portion is subtrahed from sextupla, there shall remayne tripla proportion. And thus may ye do in all others.

6. A Parallelogramme applied to a right line, is sayd to want in forme by a parallelogramme like to one geuen: whē the pa∣rallelogrāme applied wanteth to the filling of the whole line, by a parallelogramme like to one geuen:* 1.13 and then is it sayd to exceede, when it exceedeth the line by a parallelogramme like to that which was geuen.

As let E be a Parallelogrāme

[illustration]

geuen, and let AB be a right line, to whom is applied the pa∣rallelogramme ACDF. Now if it want of the filling of the line AB, by the parallelogrāme DFGB being like to the pa∣rallelogramme geuen E, then is the parallelogramme sayd to want in forme by a parallelo∣gramme like vnto a parallelogramme geuen.

Likewise if it exceede, as the parallelogramme ACGD applyed to the lin•• AB•• if it exceede it by the

[illustration]

parallelogramme FGBD being like to the parallelo∣gramme F which was ge∣uen, then is the parallelo∣gramme ABGD, sayd to exceede in forme by a pa∣rallelogramme like to a pa∣rallelogramme geuen.

This definition is added by Flussates as it seemeth, it is not in any cōmon Greke booke abroad, nor in any Commentary. It is for many Theoremes following very necessary.

Notes

• * 1.1

  The first de∣finition.
• * 1.2

  The second de••inition.
• * 1.3

  Reciprocall figures called mutuall fi∣gures.
• * 1.4

  The third de∣finition.
• * 1.5

  The fourth definition.
• * 1.6

  The fifth de∣finition.
• * 1.7

  By the name of quantities is vnderstan∣ded the deno∣minations of proportions.
• * 1.8

  Example of this definitiō.
• * 1.9

  Example in numbers.
• * 1.10

  2. 3. 15. 18.
• * 1.11

  An other example.
• * 1.12

  Of substracti∣on of propor∣tion.
• * 1.13

  The sixth de∣finition.