ex. aq. perturbato, A 12: D 9 E 4: H 3.

10°.

miscendo.

If A 12 B 6= C 6: D 3;

:

=

;

H3
by 23, V.

A 12 +B6 A 12-B 6 C6+D3:
V.

Use and App. 22,

11°.

C 6If A 5: B 4 = permutando, A 5: C 10 invertendo, C 10: A5 componendo, C 10+ A 5:

D 3.

C 10: D 8,

=

B 4: D 8, by 16, V.

D8: B 4, by B, V.

A 5 = D8 + B 4 : B 4, by 18, V. ‹

REMARKS ON BOOK V.

To the Notes and Observations gathered from various sources we simply add the commendation of BILLINGSLEY, fol. 126.

"THIS FIFTH BOOKE of EUCLIDE is of very great commoditie and vse in all Geometry, and much diligence ought to be bestowed therin. It ought of all other to be throughly and most perfectly and readily knowne. For nothyng in the bookes followyng can be vnderstood without it: the knowledge of them all depende of it.

And not onely they and other writinges of Geometry, but all other Sciences and also artes: as Musike, Astronomy, Perspective, Arithmetique, the arte of accomptes and reckoning, with other such like. This booke therefore is as it were a chiefe treasure, and a peculiar iuell much to be accompted of. It entreateth of proportion and Analogie, or proportionalitie, which pertayneth not onely vnto lines, figures, and bodies in Geometry; but also vnto soundes & voyces, of which Musike entreateth, as witnesseth Boetius and others, which write of Musike. Also the whole arte of Astronomy teacheth to measure proportions of tymes and mouings. Archimedes and Iordan, with other, writing of waightes, affirme, that there is proportion betwene waight and waight, and also betwene place and place. Ye see therefore how large is the vse of this fift booke. Wherfore the definitions also thereof are common, although here, of Euclide they be accommodate and applied onely to Geometry. The first author of this booke was, as it is affirmed of many, one Eudoxus, who was Platos scholer, but it was afterwards framed and put in order by Euclide."

GRADATIONS IN EUCLID.

BOOK VI.

THE THEORY OF PROPORTION APPLIED, FOR COMPARING THE SIDES AND AREAS OF PLANE RECTILINEAL FIGURES.

"THIS SIXTH BOOKE is for vse and practise a most speciall booke. In it are taught the proportions of one figure to an other figure, and of their sides the one to the other, and of the sides of one to the sides of an other, likwise of the angles of one to the angles of the other. Moreover it teacheth the description of figures like to figures geuen and marueilous applications of figures to lines, euenly, or with decrease or excesse, with many other theoremes, not onely of the Proportions of right lined figures, but also of sectors of circles, with their angles. On the Theoremes and Problemes of this Booke depend for the most part the compositions of all instrumentes of measuring length, breadth, or deepenes, and also the reason of the vse of the same instrumentes, as of the Geometrical square, the Scale of the Astrolabe, the quadrant, the staffe, and

such others. The vse of which instrumentes, besides all other mechanicall instrumentes of raysing up, of mouing, and drawing huge things incredible to the ignorant, and infinite other ginnes (which likewise haue their groundes out of this Booke) are of wonderfull and vnspeakeable profite, besides the inestimable pleasure which is in them."-BILLINGSLEY, fol. 153.

The Theory of Proportion, exhibited in the fifth book, is in the sixth applied to determining the proportions which exist between both the sides and the areas of similar plane rectilineal figures. The basis of the comparisons instituted is not identity of size, but identity of form; and when this cannot be predicated, or clearly inferred, no true Geometrical proportion can be established. The sixth book however advances further than this, and enables us to construct a figure, which shall possess the form of a first given figure and the size of a second. By the second book we may decribe a square equal to a given rectilineal figure;-by the sixth we may make any right-lined figure which we choose, equal in size to a given rectilineal figure.

We are also empowered, to find Lines and to draw rectilineal figures in proportion the one to the other; and to increase or diminish any figure according to a given Ratio. From this book we derive the principles of what is termed the Rule of Three, and the geometrical form for the solution of a quadratic equation; it extends also to a much wider application the fertile truth, that the square of the hypotenuse equals the sum of the squares of the sides of a right-angled triangle; and it supplies the easiest and most certain rules by which to conduct Measurements of all kinds. These will be seen when we show the Uses of various Propositions.