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Q. E. D. quod erat demonstrandum, which was the thing to be proved.
Q. E. F. quod erat faciendum, which was the thing to be done.
GRADATIONS IN EUCLID.
TREATING OF THOSE PROPERTIES OF THE CIRCLE, AND OF STRAIGHT
A circle, strictly speaking, signifies the space bounded by a circumference, but in this book the term is employed sometimes to denote that space, and at other times, the circumference itself.
Euclid, too, occasionally assumes from experimental knowledge, certain properties of the circle, which a more rigid and exact method of reasoning would have established before using them. This is the case in the first Proposition itself, where it is taken for granted that the perpendicular to the chord of an arc will meet the circle in two points. In some instances also the method of indirect demonstration is adopted, when the more satisfactory method of direct proof is available; examples of this occur in Props. 2, 13, 16 and 36.
By restricting the meaning of the term angle to an opening formed by two conterminous lines, and less than two right angles,
Euclid renders some of his demonstrations, as that of Prop. 21, more cumbersome than they need be.
The Properties of the right-angled triangle, of the circle, and of certain lines in and about a circle, as the radius, the sine, the tangent and the secant, have laid the foundations of by far the most extensive branch of Mathematics. Trigonometry, Plane and Spherical, resting on these properties and at first "confined to the solution of one general problem, has now spread its uses over the whole of the immense domains of the mathematical and physical sciences."-LARDNER's Trigonometry, p. 3.
The Learner may therefore enter on the study of this Third Book with the assurance, that he is about to cross the threshold of one of the most important parts of Plane Geometry. "The influence, indeed, of the properties of the circle upon abstract mathematical analysis has been so great that an attempt to describe the manner in which the means of expression derived from this figure has been used, would fill a volume."
The quaint English Editio Princeps of Euclid, published in 1570, thus opens to the Reader the Summary of Bk. III.
"This third booke of Euclide entreateth of the most perfect figure, which is a circle. Wherefore it is much more to be estemed then the two bookes goyng before, in which he did set forth the most simple proprieties of rightlined figures. For sciences take their dignities of the worthynes of the matter that they entreat of. But of al figures the circle is of most absolute perfection, whose proprieties and passions are here set forth, and most certainly demôstrated. Here also is entreated of right lines subtended to arkes in circles: also of angles set both at the circumference and at the centre of a circle, and of the varietie and difference of them. Wherfore the readyng of this booke, is very profitable to the attayning to the knowledge of chordes and arkes. It teacheth moreover which are circles contingêt, and
which are cutting the one the other: and also that the angle of contingence is the least of all acute rightlined angles: and that the diameter in a circle is the longest line that can be drawen in a circle. Farther in it may we learne how, three pointes beyng geuen how soever (so that they be not set in. a right line) may be drawen a circle passing by them all three. Agayne, how in a solide body, as in a Sphere, Cube, or such lyke, may be found the two opposite pointes. Whiche is a thyng very necessary and commodious, chiefly for those that shall make instrumentes seruyng to Astronomy and other artes."-BILLINGSLEY'S EUCLID, fol. 81.
1. Equal circles are those of which the diameters are equal, or, from the centres of which the straight lines to the circumferences are equal.
The criterion of the equality of circles is that their diameters or their radii are equal;-but this is neither a Definition nor an Axiom; properly it is a Theorem, the truth of which may be proved by superposition; for if centre be placed on centre and the equal radii or diameters on each other, the circumference of the one will in each point coincide with the circumference of the other, and thus the space included by one circumference will equal the space included by the
2. A straight line is said to touch a circle, i. e. is a Tangent, when it meets the circle, and being
produced does not cut it; as AB tangent to EFC in C.
The point in which the straight line meets the circle is the point of contact; the straight line "does not cut," i. e. does
pass into the circle. A Secant is a straight line which, when it meets the circle and is produced, passes into the
circle, i.e., cuts or crosses the circumference; as BHD, secant to EFC in H.