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

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.* 1.1

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.* 1.2 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.* 1.3 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,* 1.4 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.* 1.5 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 ••/••,* 1.6 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.* 1.7 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.

Notes

• * 1.1

  The fifth de∣finition.
• * 1.2

  By the name of quantities is vnderstan∣ded the deno∣minations of proportions.
• * 1.3

  Example of this definitiō.
• * 1.4

  Example in numbers.
• * 1.5

  2. 3. 15. 18.
• * 1.6

  An other example.
• * 1.7

  Of substracti∣on of propor∣tion.