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

A Corollary. 2.

It is also manifest, that you may by any of the fower wayes, here specified, proceede infinitely in the proportion of a line deuided by extreame & middle proportion:* 1.1 And in the first, and fourth wayes, en∣creasing continually the quantities of the lin••s made: but in the second and third wayes, dim••nishing continually the quantities of the sayd whole lines, made (and thereby their segmentes). And ye••, ne∣uerthelesse reteyning in euery line made (by ••••y of the wayes) and in his segments all, and the same properties, which the first line, and his segment •••• haue. After which ••are of Progression, as the termes in continuall proportion do encrease, and are m••e in number: So, likewise, do the middle proportionalls, (accordingly) become moe: But euer fower in number, by two, then the termes of the Progression are.

¶What Resolution is.

Resolution, is the assumption or taking of the thing which is to be proued, as graunted, and by thinges which necessarily follow it,* 1.2 to passe vnto s••me truth graunted.

¶ What Composition is.

Composition, is an assumption or taking of a thing graunted, and by thinges which of necessity follow it, to passe vnto the finding out of the thing sought or to be proued.

Resolution of the first Theoreme.

Suppose that a certaine right line AB, be diuided by an extreame & meane proportion in the point C, & let the greater segmēt therof be AC, vnto which adde a line equal to the halfe of the line AB, and let that line be AD. Then I say that the square of the line CD is quin∣tuple to the square of AD. For forasmuch as the square of the line CD is quintuple to the square of AD: but the square of the line

[illustration]

CD is (by the 4. of the second) equall to that which is composed of the squares of the lines CA, & AD, together with that

which is contayned vnder the lines CA, and AD twise. Wh••••f••re that which is composed of the squares of the lines CA, & AD, together with that which is cōtained vnder the lines C∣A, & AD twise is quintuple to the square of the line AD. Wherfore, that which is composed of the square of the line CA together with that, which is contayned vnder the lines CA, and AD twise is quadruple to the square of the line AD. But vnto that, which is contayned vnder the lines CA, and AD, twise, is equall that which is contayned vnder the lines CA, and AB, for the line AB is double to the line AD. And vnto the square of the line A∣C, is equall that which is cōtayned vnder the lines AB & BC, for the line AB, is by supposi∣tion diuided by an extreme and meane proportion, in the point C. Wherefore, that, which is contayned vnder the lines AB, and AC, together with that which is contayned vnder the lines AB, and BC, is quadruple to the square of the line AD. But that, which is composed of that which is contayned vnder the lines AB, and AC, together with that which is contay∣ned vnder the lines AB, and BC, is the square of the line AB (by the 2. of the second.) Wher∣fore the square of the line AB is quadruple to the square of the line AD. And so is it in deede: for the line AB is double to the line AD: as was at the first supposed.

Composition of the first Theoreme.

Now forasmuch as the square of the line AB is quadruple to the square of the line AD, but the square of the line AB, is that which is contayned vnder the lines AB, and AC, toge∣ther with that, which is contayned vnder the lines BA, and BC. Wherefore that which is contayned vnder the lines BA, and AC, together with that which is contayned vnder the lines BA, and BC is quadruple to the square of the line AD. But that which is contayned vnder the lines BA, and AC, is equall to that which is contayned vnder the lines DA, and AC twise (by the 1. of the sixth), and that which is contayned vnder the lines AB, and ••C is equall to the square of the line AC, by the definition of a line diuided by extreme and meane proportion. Wherefore the square of the line AC, together with that, which is contay∣ned vnder the lines DA, and AC twise, is quaduple to the square of the line DA. Wherfore that which is composed of the squares of the lines DA, and AC, together with that which is contayned vnder the lines DA, and AC, twise, is quintuple to the square of the line DA. But that which is composed of the squares of the lines DA, and AC, together with that which is contayned vnder the lines DA, and AC twise, is equall to the square of the line CD (by the 4. of the second). Wherefore the square of the line CD is quintuple to the square of the line AD: which was required to be demonstrated.

Resolution of the 2. Theoreme.

Suppose that a certayne right line, CD, be quintuple to a segmēt of the same line, namely, to DA: and let the double of the line DA, be AB. Thē I say that the line AB is diuided by an extreme and meane proportion in the point C•• and the greater segmēt therof is AC, which is the rest of the right line put at the beginning. For forasmuch as the line AB is diuided by an extreame and meane proportion in the poynt C, and the greater segment thereof is the line AC, therefore that which is contained vnder the lines A••, and BC, is equall to the square of the line AC. But that which is contay∣ned

[illustration]

vnder the lines BA, and AC, is e∣qual to that which is contayned vn∣der the lines DA, and AC twise•• for the line BA, is double to the line AD. Wherefore that which is contayned vnder the lines AB, and BC together with that which is cōtayned vnder the lines BA, and AC, which is the square of the line AB (by the 2. of the second) is equall to

that which is contayned vnder y^e lines DA, & AC, twise together with the square of the line AC. But the square of the line AB, is quadruple to the square of the line DA. Wherfore that which is contayned vnder the lines DA, and AC, twise, together with the square of the line AC, is quadruple to the square of the line AD. Wherefore the squares of the lines DA, and AC, together with that which is contayned vnder the lines DA, and AC, twise, which is the square of the line DC, are quintuple to the square of the line DA. And so are they in deede by supposition.

Composition of the 2. Theoreme.

Now forasmuch as the square of the line CD, is quintuple to the square of the line DA•• But the square of the line CD, is that which is cōposed of the squares of y^e lines DA, & AC, together with that which is cōtained vnder the lines DA, & AC, twise: Wherfore y^e squares of the line DA, & AC together with that which is cōtayned vnder the lines DA, & AC, twise, are quintuple to the square of the line DA. Wherfore, by diuisiō, that which is cōtained vnder the lines DA, and AC, twise together with the square of the line CA, is quadruple to the square of the line AD. And the square of the line AB, is quadruple to the square of the line AD. Wherefore that which is contayned vnder the lines DA, and AC twise, which is that, which is contayned vnder the lines BA, and AC once, together with the square of the line AC, is equall to the square of the line AB. But the square of BA, is that which is contayned vnder the lines BA, and AC, together with that which is contayned vnder the lines BA, and BC. Wherfore that which is contayned vnder the lines BA, & A∣C, together with that which is cōtayned vnder the lines AB, & BC, is equall to that which is contayned vnder the lines BA, and AC together with the square of the line AC. Now then taking away that which is common to them both, namely, that which is cōtayned vnde•• the lines BA, and AC, the residue, namely, that which is contayned vnder the lines AB, BC is equall to the square of the line AC. Wherefore as the line BA, is to the line AC, so is the line AC to the line CB. But the line BA is greater then the line AC, wherefore the line AC also is greater then the line CB. Wherefore the line AB is diuided by an ex∣treame, and meane proportion in the poynt C, and the greater segment thereof is the line AC, which was required to be demonstrated.

Resolution of the 3. Theoreme.

Suppose that a certayne right line AB, be diuided by an extreame, and meane propor∣tion in the point C•• and let the greater segment thereof be the line AC, and let the halfe of the line AC, be the line CD. Then I say that the square of the BD is quintuple to the square of the line CD. For forasmuch as the square of the line BD, is quintuple to the square of the line CD. But the square of the line DB, is that which is contayned vnder the lines AB, and BC, together with the square of the line DC (by the 6. of the second). Wherefore that which is contayned vnder the lines AB, and BC,

[illustration]

together with the square of the line DC, i•• quintuple to the square of the line DC. Wherefore, that which is contayned vnder the lines AB, and BC, is quadruple to the square of the line DC. But vnto that which is contayned vnder the lines AB, and BC, is equall the square of the line AC: for the line AB, is diuided by an extreame and meane proportion in the point C. Wherefore the square of the line AC is quadruple to the square of the line DC: and so is it in deede, for the line AC is double to the line DC.

Composition of the 3. Theoreme.

Forasmuch as the line AC is double to the line DC, therefore the square of the line AC is quadruple to the square of the line DC (by the 20. of the sixth). But vnto the square of the line AC, is equall that which is contayned vnder the lines AB, and BC, by supposi∣tion: wherefore that which is contayned vnder the lines AB, and BC, is quadruple to the square of the line CD. Wherefore, that which is contayned vnder AB, and BC, to∣ther with the square the line DC, which is the square of the line DB (by the 6. of the se∣cond) is quintuple to the square of the line DC: which was required to be demonstrated.

Resolution of the 4. Theoreme.

Suppose that a certayne right line AB, be diuided by an extreme and meane proportion in the point C. And let the greater segment thereof be AC. Then I say that the squares of the lines AB, and BC, are treble to the square of the line AC. For forasmuch as the squares of the lines AB, and BC, are treble to the square of the line AC, but the squares of the lines AB, and BC, are that which is contayned vnder AB, and BC, twise together with the square of the line AC (by the 7. of the second). Wherefore that which is contayned vnder the lines AB, and BC, twise, together with the square

[illustration]

of the line AC, is treble to the square of the line AC. Wherefore, that which is contayned vnder the lines AB, & BC, twise, is double to the square of the line AC. Wherefore that which is contayned vnder the lines AB, and BC, once, is equall to the square of the line AC. And so it is in deede. For the line AB is diuided by an extreme, and meane proportion in the point C.

Composition of the 4. Theoreme.

Forasmuch therefore as the line AB, is diuided by an extreme and meane proportion in the poynt C, and the greater segment thereof is the line AC•• therfore that which is contay∣ned vnder the lines AB, and BC is equall to the square of the line AC. Wherfore that which is cōtayned vnder the lines AB, and BC twise is double to the square of AC. Wherfore that which is contayned vnder the lines AB, and BC, twise, together with the square of the line AC, is treble to the square of the line AC. But that which is contayned vnder the lines A∣B, and BC, twise, together with the square of the line AC, is the squares of the lines AB, and BC (by the 7. of the second). Wherefore the squares of the lines AB, and BC, are treble to the square of the line AC: which was required to be demonstrated.

Resolution of the 5. Theoreme.

Suppose that a certaine right line AB, be diuided by an extreme and meane proportion in the point C. And let the greater segment therof be the line AC. And vnto the line AB, adde a line equall to the line AC, and let the same be AD. The•• I say that the line D∣B, is diuided by an extreme and meane proportion in the point A. And the greater segment therof is the line AB. For forasmuch as the line DB is diuided by an extreme & meane pro∣portion in the point A, and the greater segment thereof is the line AB, therfore as the line DB, is to the line BA, so is the line B∣A,

[illustration]

to the line AD: but the line AD, is equall to the line AC: wherefore as

the line DB, is to the line BA, so is the line BA to the line AC. Wherfore by conuersiō as the line BD is to the line DA, so is the line AB to the line BC (by the corollary of the 19. of the fifth): wherfore by diuision, by the 17. of the fifth, as the line BA, is to the line AD, ••o is the line AC, to the line CB. But the line AD is equall to the line AC. Wherfore as the line BA, is to the line AC, so is the line AC to the line CB. And so it is indeede, for the line AB is, by supposition, diuided by an extreme and meane proportion in the point C.

Composition of the 5. Theoreme.

Now forasmuch as the line AB, is diuided by an extreme and meane proportion in the point C: therefore as the line BA is to the line AC, so is the line AC to the line CB: but the line AC is equall to the line AD. Wherefore as the line BA is to the line AD, so is the line AC to the line CB. Wherfore by composition (by the 18. of the fifth) as the line BD is to the line DA, so is the line AB to the line BC. Wherefore by conuersion (by the corollary of the 19. of the fiueth) as the line DB is to the line BA, so is the line BA to the line AC: but the line AC is equall to the line AD. Wherefore as the line DB is to the line BA, so is the line BA to the line AC. Wherfore the line DB, is deuided by an extreme and meane proportion in the point A: and his greater segment is the line AB: which was required to be demon∣strated.

An Aduise, by Iohn Dee, added.

SEing, it is doubteles, that this parcel of Resolution and Composition, is not of Euclides doyng: it can not ••ustly be imputed to Euclide, that he hath, therby, eyther superfluitie or any part disproportioned in his whole Composition Elementall. And though, for one thing, one good demonstration well suffiseth: for stablishing of the veritie: yet, o•• one thing diuersly demonstrated: to the diligent examiner of the diuerse meanes, by which, that varietie ariseth, doth grow good occasions of inuenting demon∣strations, where matter is more straunge, harde, and barren. Also, though resolution were not in all Euclide before vsed: yet thankes are to be geuen to the Greke Scholic writter, who did leaue both the definition, and also, so short and easy examples of a Method, so auncient, and so profitable. The antiquity of it, is aboue 2000. yeares: it is to we••e, euer since Plato his time, and the profite, therof so great, that thus I finde in the Greeke recorded.* 1.3 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. Proclus hauing spoken of some by nature, excellent in inuenting demonstrations, pithy and breif sayeth:

Yet are there Methods geuen [for that purpose]. And in dede, that, the best, which, by Resolution, reduceth the thing inquired of, to an vndoubted principle. Which Method, Plato, taught Leodamas (as i•• re∣ported)

And he is registred, thereby, to haue bene the inuenter of many things in Geometry.

And, verely, in Problemes, it is the chief ayde for winning and ordring a demonstration: first by Supposition, of the thing inquired of, to be done: by due and orderly Resolution to bring it to a stay, at an vndoubted veritie. In which point of Art, great abundance of examples, are to be seen, in that excellent and mighty Mathematici••n, Archimedes: & in his expositor, Eutocius, in Menaechmus like∣wise: and in Diocles booke, de Pyti••s: and in many other. And now, for as much as, our Euclide in the last six Propositions of this thirtenth booke propoundeth, and concludeth those Problemes, which were the ende, Scope, and principall purpose, to which all the premisses of the 12. bookes, and the rest of this thirtenth, are directed and ordered: It shall be artificially done, and to a great commodity, by Resolution, backward, from these 6. Problemes, to returne to the first definition of the first booke: I meane, to the definition of a point. Which, is nothing hard to do. And I do counsaile all such, as desire to attein••, to the profound knowledge of Geometrie, Arithmeticke, or any braunche of the sciences Mathematicall, so by Resolution, (discreatly and aduisedly) to resolue, vnlose, vnioynt and disseauer euery part of any worke Mathematicall, that, therby•• aswell, the due placing of euery verity, and his proofe: as also, what is either superfluous, or wanting, may euidently appeare. For so to inuent, & there with to order their writings, was the custome of them, who in the old time, were most excellent. And I (for my part) in writing any Mathematicall conclusion, which requireth great discourse, at length haue found, (by experience) the commoditie of it, such: that to do other wayes, were to me a confu∣sion, and an vnmethodicall heaping of matter together: besides the difficulty of inuenting the matter to be disposed and ordred. I haue occasion, thus to geue you friendely aduise, for your be••ofe•• because some, of late, haue inueyed against Euclide, or Theon in this place, otherwise than I would wish they had.

Notes

• * 1.1

  Note 4. wayes of progres••ion, in the proportion of a line deuided by extreme and middle propor∣tion.
• * 1.2

  What resolution and composition is, hath before bene taught in the beginning of the first booke.
• * 1.3

  Proclus in the Greeke: in the 58. page.