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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 not 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.
The terms Tangent and Secant are often restricted in meaning;—the first to the line which by one extremity touches an extremity of the diameter at right angles to it, and which has its other extremity terminated by a straight line, the Secant, from the centre of the circle across the circumference; the second to the line from the centre, across the circumference, and terminated by the tangent. Thus BC is the tangent and DB the secant of the arc CH, or of the angle CDB measured by the arc; the name Cosine being given to the space DK cut off between the centre D and the sine; and Versed Sine to the space KC between the sine and the tangent point C.
The term Sine denotes a perpendicular to a diameter from the point where the secant crosses the circumference; as HK.
The terms Tangent, Secant, Sine, &c., thus restricted, were of continual use in Trigonometry; and with a widely extended meaning are now constantly employed. The process is instructive, by which extension has been given to Trigonometrical Symbols, and may thus be briefly stated; 1. Sine, cosine, &c., at first denoted lines so named drawn in and about a circle, with reference to an angle at the centre, and measured by its arc, each angle having a different sine, &c., according as the radius of the circle was increased or diminished in length.
2. To avoid these continual diversities, that radius was supposed always to be a unit, or rather the unit of measurement for the other lines; and the secant, sines, &c., to be multiples or fractional parts of that unit; thus the sine of 60°, being equal to the radius, was unity, or 1, and the sine of 30° was, or 5. The names sines, cosines, &c., in this way lost their first meaning; they denoted, not lines, but the numerical ratios of those lines to the radius, and were abstract numbers.
3. Another step was to represent the angle itself by an abstract number. Degrees and minutes had been the measure of the central angle, that angle was measured by its arc, and the arc bore a numerical ratio to the unit of measurement, the radius.
4. A fourth step made the process perfect. Hitherto the sum of the angles could not exceed four right angles, but this limit also was to be passed. The idea of a line revolving round a point, and continuing its rotation after a revolution had been completed, originated the method of using angles consisting of more than four right angles.
Thus angles, sines, &c., were all represented by numbers; and though the old names were retained, Trigonometry which at first was a simple application of Geometrical truths, and which still rests on Geometry for its foundation, became a branch of the higher Arithmetic, and has its operations conducted on arithmetical and algebraical principles.
3. Circles are said to touch one another, which meet, but do not cut one another. Thus the circle of which L is the centre, touches EFC in E, and circle G touches it in F.
4. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal; thus EF and GH are equally distant from C, when perp. CA = perp. CB.
5. And the straight line on which the greater perpendicular falls, is said to be farther from the centre; thus IK is farther from C than G H is, because the perp. CD > perp. C B.
As the distance from the vertex of a triangle to its base is measured by a perpendicular, so the distance of a straight line from the centre of a circle is the perpendicular drawn to it from the centre. A Proposition analagous to Props. 7 and 8, Book III., would explain "why the perpendicular from a point on a straight line is called the distance from that line."
6. A segment of a circle is the figure contained by a straight line, and the circumference which it cuts off; as the fig. ABCA.
"A figure included by an arc and its chord is
The straight line of a segment, as AB, is named
7. The angle of a segment is that which is contained by the straight line and the circumference; as angle ABC contained by AB and the arc BCA.
8. An angle in a segment is the angle contained by two straight lines drawn from any point of the circumference of the segment, to the extremities of the straight line which is the base of the segment; as ACB, or ▲ ADB.