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A Lesser quantity compared with a greater, is called a part.
The whole corresponds to its part: and this shall be the greater quantity compared with the lesser; whether it contains the same in effect, or that it doth not contain the same.
Parts or quantities taken in general are
divided ordinarily into Aliquot parts, and Aliquant parts.
1. An Aliquot part (which Euclid defines in this Book) is a Magnitude of a Magnitude, the lesser of the greater, when it measureth it exactly. That is to say, that it is a lesser quantity compared with a greater, which it measureth precisely. As the Line of Two Foot taken Three times, is equal to a Line of Six Foot.
2. Multiplex is a Magnitude of a Magnitude, the greater of the lesser, when the lesser measureth the greater: That is to say, that Multiplex is a great quantity compared with a lesser, which it contains precisely some number of times. For Example, the Line of Six Foot, is treble to a Line of Two Foot.
Aliquant parts, is a lesser quantity compared with a greater, which it mea∣sureth not exactly. So a Line of 4 Foot, is an Aliquant part of a Line of 10 Foot.
Equimultiplexes are Magnitudes which contain equally their Aliquot parts, that is to say the same number of times,
| 12. | 4. | 6. | 2. |
| A | B | C | D |
3. Reason, (or Ratio) is a mutual habitude or respect of one Magnitude to another of the same Species. I have added of the same Species.
4. For Euclid saith, that Magnitudes have the same reason, when being multi∣plied, they may surpass each other. To do which, they must be of the same Species. In effect, a Line hath no manner of Reason with a Surface, because a Line taken Ma∣thematically is considered without any Breadth: so that if it be multiplyed as many times as you please, it giveth no Breadth, and notwithstanding a Surface hath Breadth.
Seeing that Reason is a mutal habitude or respect of a Magnitude to another, it ought to have two terms. That which the Philo∣sophers would call foundation, is named by the Mathematicians Antecedent, and the term is called Consequent. As if we com∣pare the Magnitude A, to the Magnitude I, this habitude or Reason shall have for Antecedent the quantity A, and for conse∣quent the quantity B. As on the contrary, if we compare the Magnitude B, with A, this Reason of B to A, shall have for Ante∣cedent the Magnitude B, and for consequent the Magnitude A.
The Reason or habitude of one Magni∣tude to another, is divided into rational Reason, and irrational Reason. Rational Reason is a habitude of one Magnitude to another, which is commensurable thereto; that is to say, that those Magnitudes have a common measure, which measureth both exactly. As the reason of a Line of 4 Foot, to a Line of 6, is rational; because a Line of two Foot measureth both exactly: and when this happeneth, those Magnitudes have the same Reason as one Number hath to another. For Example, because that the Line of two Foot, which is the common measure, is found twice in the Line of 4 Foot, and thrice in the Line of 6; the first to the second shall have the same Reason as 2 to 3.
Irrational Reason is between Two Mag∣nitudes of the same Species which are in∣commensurable, that is to say, that have not a common measure. As the Reason of the Side of a Square to its Diagonal. For there cannot be found any measure, although never so little, which will measure both precisely.
Four Magnitudes shall be in the same Reason, or shall be Proportionals, when the Reason of the first to the second, shall be the same, or like to that of the third to the
fourth: wherefore to speak properly, Pro∣portion is a similitude of Reason. But one findeth it difficult to understand in what consisteth this similitude of Reason. It is only to say that two habitudes or Relations be alike. For Euclid hath not given a just Definition, and which might have explained its Nature; having contented himself to give us a mark by which we may know, if Magnitude have the same Reason. And the obscurity of this Definition hath made this Book difficult. I will endeavour to supply this default.
5. Euclide saith, that Four Magnitudes have the same Reason, when having taken the Equi-multiplices of the first, and of the third; and other Equi∣multiplices of the second, and of the fourth; whatever combination is made, when the Multiplex of the first is greater than the Multiplex of the second; the Multiplex of the third shall be also grea∣ter than the Multiplex of the fourth: And when the Multiplex of the first is equal, or less than the Multiplex of the second; the Multiplex of the third is equal or less than the Multiplex of the fourth. That then there is the same Rea∣son between the first and second; as there is between the third and fourth.
| A | B | C | D |
| 2. | 4. | 3. | 6. |
| E | F | G | H |
| 10 | 8. | 15 | 12. |
| K | L | M | N |
| 8. | 8. | 12. | 12. |
| O | P | Q | R |
| 6. | 16 | 9. | 24 |
To explain well what Proportion is, it is to say, that four Magnitudes have the same Ratio; although one may say in general, that to that end the first must be alike part, or a like whole, in respect of the second; as is the third, compared to the fourth: notwithstanding because this Definition doth not convene with the Reason of equality, there must be given a more general; and to
make it intelligible, it must be explained what is meant by a like Aliquot part.
Like Aliquot parts are those which are as many times in their whole, as three in respect of nine; two in respect of six, are alike Aliquot parts, because each are found three times in their whole.
The first quantity will have the same Reason to the second, as the third hath to the fourth, if the first contains as many times any Aliquot part of the second whatever, as the 3d. contains alike Aliquot part of the 4th.
| A, | B, | C, | D. |
To make this Definition yet clearer; I will in the first place prove, that if there be the same reason of A to B, as there is of C to D; A will contain as many times the Aliquot parts of B, as C doth of D. And I will afterwards prove, that if A contains as many times the Aliquot parts of B, as C doth of D; there will be the same Reason of A to B, as of C to D.
The first Point seemeth evident enough, provided one doth conceive the terms: for if A contains one Hundred, and one times the tenth part of B; and C only One Hundred times the tenth part of D: the Magnitude A compared with B, would be a greater whole, than C compared with D: so that it could not be compared after the same manner, that is to say, the habitude or Relation would not be the same.
The second point seemeth more difficult; to wit, whether if this propriety be so found, the Reason shall be the same, that is to say, if AB contains as many times any Aliquot parts whatever of CD, as E contains like Aliquot parts of F: there shall be the same Reason of AB to CD, as of E to F. For I will prove that if there were not the same Reason, A would contain more times any Aliquot part of B, than C containeth alike Aliquot parts of D: which would be con∣trary to what we had supposed.
Let us suppose then that there is a greater reason of AB to CD, than of E to F;
a lesser Magnitude than AB, for Example, AG shall have the same Reason to CD, as E to F. Divide CD in the middle in H and HD, in the middle in I and ID, in the middle in K. In continu∣ing this Division, you would find an Aliquot part of CD less than GB; let it be KD.
Demonstration. Seeing there is the same Reason of AG to CD, as of E to F; AG will contain as many times KD an Aliquot part of CD, as E would contain a like Aliquot part of F. Now AB con∣tains KD, once more than AG: thence AB will contain once more KD an Aliquot part of CD, than E doth contain a like Aliquot part of F; which would be contrary to the supposition.
6. There will be a greater Reason of the first quantity to the second, than of the third to the fourth: if the first con∣tains more times any Aliquot part of the second, than the third doth contain a like Aliquot part of the fourth. As 101 hath a greater Reason to 10 than 200 to 20; because that 101 contains One Hun∣dred and one times the Tenth part of 10, and 200 contains only One Hundred times the Tenth part of 20, which is 2.
7. The Magnitudes or quantities which are in the same Reason, are called Pro∣portionals.
8. A Proportion or Analogie, is a Similitude of Reason or habitude.
9. A Proportion ought to have at least three terms. For to the end there be simi∣litude of Reason, there must be two Reasons: Now each Reason having two terms, the antecedent and the consequent, it seemeth there ought to be four, as when we say that there is the same Reason of A to B, as of C to D: but because the consequent of the first Reason may be taken for antecedent in the second, three terms may suffice, as when I say that there is the same Reason of A to B, as of B to C.
10. Magnitudes are in continued Pro∣portion, when the Terms between them are taken twice; that is to say, as an∣tecedent and as consequent. As if there be the same Reason of A to B, as of B to C, and of C to D.
11. Then A to C shall be in du∣plicate Ratio of A to B: and the Ratio of A to D shall be in triplicate Ratio to that of A to B.
It is to be taken notice, that there is a great deal of difference between double Ratio and duplicate Ratio. We say that the Ratio of four to two is double, that is to say, four is the double of two, whence it followeth that the number two, is that which giveth the Name to this Ratio, or rather to the Antecedent of this Ratio. So we we say, double, triple, quadruple, quintuple, which are Denominations taken from those numbers, duo, tres, quatuor, quinque, compared with unity: for we better conceive a Reason when its terms are small. But as I have already taken notice, those De∣nominations fall rather on the Antecedent than on the Reason it self: we call that double, triple, Reason or Ratio, when the Antecedent is double or triple to the con∣sequent: but when we say the Reason is duplicate, we mean a Reason compounded of two like Reasons, as if there be the same Reason of two to four, as of four to eight; the Reason of two and eight being com∣pounded of the Reason of two and four, and of that of four and eight, which are alike; and as equal the Reason or Ratio of two to eight, shall be duplicated by each. Three to twenty seven, is a duplicated Reason of that of three to nine. The Reason of two to four is called subduple;
that is to say, two is the half of four: but the reason of two to eight is duple of the sub-duple; that is to say, that two is the half of the half of eight: as three is the third of the third of twenty seven; where you see there is taken twice the Denominator ½ and ⅓. In like manner, eight to two is a duplicate reason of eight to four, because eight is double to four, but eight is the double of the double of two. If there be four terms in the same continued Reason, that of the first and last is triple to that of the first and second, as if one put these four Numbers, two, four, eight, sixteen; the reason of two to sixteen is triple of two to four; for two is the half of the half of the half of sixteen. As the reason of sixteen to two, is triple of sixteen to eight; for sixteen is the double of eight, and it is the double of the double of the double of two.
12. Magnitudes are homologous, the Antecedents to the Antecedents, and the Consequents to the Consequents. As if there be the same Reason of A to B, as of C to D, A and C are homologous, or Magnitudes of a like Ratio.
The following Definitions are ways of arguing by Proportion, and it is princi∣pally to demonstrate the same that this Book is composed.
13. Alternate Reason, or by Permu∣tation, or Exchange, is, when we com∣pare the Antecedents one with the o∣ther, as also the consequents. For ex∣ample, if because there is the same reason of A to B as of C to D, I conclude there is the same reason of A to C as of B to D; this way of reasoning cannot take place but when the four terms are of the same Specie; that is to say, either all four Lines, or Superficies, or Solids. Proposition 16.
14. Converse, or Inverse Reason, is a comparison of the Consequents to the Antecedents. As, if because there is the same reason of A to B as of C to D; I conclude, there is the same reason of B to A as there is of D to C. Propositi∣on 16.
15. Composition of Reason is a com∣parison of the Antecedent and Conse∣quent taken together, to the Consequent alone. As, if there be the same Reason of A to B as of C to D; I conclude also, that there is the same reason of AB to B as of CD to D. Prop. 18.
16. Division of Reason is a compa∣rison of the excess of the Antecedent above the Consequent, to the same Con∣sequent. As, if there be the same reason of AB to B, as of CD to D; I conclude
that there is the same reason of A to B, as of CD. Prop. 17.
17. Conversion of Reason is the com∣parison of the Antecedent to the diffe∣rence of the Terms. As, if there be the same reason of AB to B, as of CD to D; I conclude, that there is the same reason of AB to A, as of CD to C. Propositi∣on 18.
18. Proportion of Equality is a com∣parison of the extream Quantities in leaving out those in the middle.
| A, | B, | C, | D, |
| E, | F, | G, | H. |
19. Proportion of Equality well rank∣ed, is that in which one compareth the Terms, in the same manner of Order; as in the preceding Example. Prop. 22.
20. Proportion of Equality ill ranked, is that in which one compareth the Terms with a different Order. As if there were the same reason of A to B as of G to H, and of B to C as of F to G, and of C to D as of E to F. I draw this Conclusion, that there is the same reason of A to D as of E to H. Prop. 28.
Here is all the ways of arguing by Pro∣portion. There is the same reason of A to B as of C to D; therefore by alternate reason there is the same reason of A to C as of B to D: and by inversed reason, there is the same Reason of B to A as of D to C; and by composition, there is the same reason of AB to B as of CD to D.
By Division of Reason, if there be the same reason of AB to B as of CD to D, there is the same reason of A to B as of C to D; and by Conversion, there is the same reason of AB to A as of CD to C.
By reason of Equality well ranked, if there be the same reason of A to B as of C to D; and also the same reason of B to E as of D to F; there will be the same rea∣son of A to E as of C to F.
By reason of Equality ill ranked, if there be the same reason of A to B as of D to F, and also the same reason of B to E as of C to D: there will be the same reason of A to E as of C to F.
This Book contains twenty five Propositi∣ons of Euclid, to which there has been ad∣ded ten, which are received. The first six of this Book are useful only to prove the fol∣lowing Propositions by the method of Equi-multiplices: and seeing I do not make use of that method, I begin at the seventh,
without changing the Order or Number of the Propositions.