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ilaring this pointed out some of the propezues of bi 11.- fitum, the next schject of consideration as be it ose oi curve 'ined figures, particularly ci the circle.
The characteristic of a circle is, that esery part ciis erteti, called the circumferie or garipety, is equals disan! from a point within it calles the cerire.
Every right line drawn from the centre to the circumference, is called a radius, as CA, CB, CH, and CD, in fig. 22; and any right line passing through the centre, and terminated both ways by the circumference, is called a diameter, 25 ACD, which is double the length of the radius, for AC being equal to CD, the whole AC must be double either of the parts AC or CD.
All straight lines drawn in a circle and terminated at each end by the circumference, but not passing through the centre, are terned chords ; such as AG, and HD.
An arch or arc is any portion of the circumference of a circle; as the small portion AG, or the great portion ABUDG: the curve line HD is also an arch of the circle.
A figure contained between an arch of a circle and the chord or line joining the extremities of the arch, is called a segment of a circle: thus, the figure comprehended between the curve line AG and the chord AG is a segment; and, on the other hand, the figure contained within the same chord AG and the great curve ABHDG, is also a segment.
If the figure be formed by a line passing through the centre, that is, by a diameter, the segments are equal to one another, each being one half of the circle, or a semicircle, as ABD and AGD. A figure contained within an arch of a circle, as BH, the
radius BC, and the radius HC, is called a sector. HCD, and ACH, are also sectors; but if the one radius be perpendicular to the other, that is, if they form a right angle at the centre, as BC and DC, then the sector is called a quadrant, as being the fourth part of the whole circle, and the half of a semicircle; for AD, the diameter, being equal. ly divided in the centre C, and CB being perpendicular to the diameter at that point, any point in CB will be equally distant from A and B; the arch AB will therefore be equal to the arch BD, and the segment ABC to the segment BDC; each of them will, consequently, be one half of the semicircle ABD, that is, each will be one fourth of the whole circle, or a quadrant.
A right line can meet the circumference of a circle only in two points, as AG or HD.
In the same circle, or in circles equal to each other, the chords of equal arches are equal to each other; and vice versa, the arches subtended by equal chords are equal to each other.
In the same or in equal circles, the greater arch is subtended by the greater chord, and the less arch is subtended by the less chord ; unless the arch be greater than a semicircle, when the greater the arch the smaller the chord.
Prop. IX. fig. 23. If three points, as ABD, be taken in the circumference of a circle whose centre is C, no other circle can be drawn through the same three points not coinciding with the given circle. Join the three points by the two chords AB and BD, and from the centre C draw CE and CF perpendicular to these chords, and consequently bisecting them, or dividing them severally into two equal parts ; in which case the centre of any circle passing through the points A and B, will be somewhere in the line EC; and in the same way the centre of any circle passing through the points B and D must be somewhere in the line 3 A 2
FC: 0 ite92e2ty the centre of a circit passin lugt laze turee points AB and D I s1 be ubare best ins 11:45 EC and FC intri, that is, in the per C: but this is the certre of the circit in ubos circunferenee ate three gre poi:15 were taken; therefore orly oge circle can be CEx10 Fass 11.19 gli ang three giren poats: and bener i foss, thai circles can cut one another only in two pom's.
Psor. X. £g. 24. The angle frossed to the lines AC aud BC drawn from the extremities of an arch of a circle A? 10 the centre C, is doable the argje formed by its lines AD and BD drawn from the same entre les 10 any posi, as D in the cpposite circumference. Fmn D draw through the centre C the diameter DE. In the triangle DCA ibe sides DC and CA are equal to each orbes, each being a radio:s of the circle; the atgles opposite to them are therefcre equal, t.at is, CAD is equal to CDA : but, by the 7th Prop, it appears, that the exterior angle A CE, firmed by producing the side DC, is equal to the two inward and opporile angles of the triangle, that is, sirce they are equal to one another, ACE must be double ADC or ADE; again, in the triangle CDB, by the same Proposition, it appearö that the exterior angle ECB is equal to the two inward and opposite angles CDB and CBD; but these angles being opposite to equal sides, must be also equal; and the exterior angle ECB will be double CDB, that is, EDB: it follows, therefore, that the whole angle at the centre ACB, will be double the whole angle at the circumference ADB.
Corollary. All angles, however situated, of the same segment of a circle, or in other words, all angles formed by liues drawn from the extremities of the arch of the segment, and meeting in the circumference, are equal.
From this proposition it follows, that the angles in a semicircle are all right angles; that the angles in a segment
greater than a semicircle are less than a right angle; that those in a segment less than a semicircle are greater than a right angle; and that the opposite angles of any quadrilateral figure, drawn within and bounded by the circumference of a circle, are together equal to two right angles.
When two or more figures have their respective angles equal and their respective sides proportionals, these figures are said to be similar : and these respective angles and sides are called homologous.
When two figures, being applied the one to the other, perfectly coincide in every part, they are said to be equal.
When two figures, whatever be their shape, contain surfaces of equal extent, they are said to be equivalent.
Two equal figures must be similar, but two similar figures may be very unequal.
PROP. XI, fig. 25. · Parallelograms situated on the same or equal bases, and between the same parallels, or of equal altitudes, are equivalent to one another : thus the parallelograms ABDC and AEFB being situated on the same base AB, and being of the same supposed altitude, the sides CD and EF opposite to the base will be in the same parallel CDEF: and from the definition formerly given of a parallelogram, the opposite sides will be respectively equal to one another; AC will be equal to DB, and AE to BF: again, CD and E being both equal to AB, they will be equal to one anoth r. To these equals add the line DE, and we shall have he whole CE equal to the whole DF; and consequently tie triangles CEA and DFB with three sides of the one equal to the corresponding three sides of the other; these triangles will therefore be equal. (See Prop. 3.) If now from the quadrilateral ACFB we take away the triangle CEA, we have remaining the parallelogram AEFB; and if from the same quadrilateral we take away the triangle DFB, we have remaining the parallelogram
ACDB: but if from equal quantities we take away equal quantities, the remainders will be equal ; we have consequently the parallelogram · ACDB equal to the parallelo
Hence it follow3, that every parallelogram is equivalent to a rectangle of equal base and altitude, as in this example, when ACDB is a right-angled parallelogram.
In the same way it may be proved that the parallelograms AEFB and CEFD, in fig. 26, are equivalent to each other; for, by the nature of the parallelograms, the opposite sides being respectively equal, the line AB will be equal to the line CD: if from these equals we take away CB, which is common to both, we shall have the part AC equal to the part BD, and consequently the triangle AEC equal to BFD; if, therefore, from the quadrilateral AEFD, we take away the triangle AEC, we have the parallelogram CEFD; and if from the same quadrilateral we take away the triangle BFD, we shall have the parallelogram AEFB, which, from what was before said, must be equal to the parallelogram CEFD.
Prop. XII. fig. 27. Every triangle, as ABC, is one half of the parallelogram ABDC, situated on the same base AC, and of the same altitude BG; for the base AC is equal to BD, the side AB to CD, and the side BC is common to both triangles, which, by Prop. 3, are therefore equal : but these equal triangles form the parallelogram ABDC : the triangle ABC is consequently one half of the parallelo
PROP. XIII. fig. 28. Two right-angled parallelograms, of equal altitudes, are to one another in the proportion of their bases. Let the two rectangles, ABCD and DCEF, have the same altitude DC, these rectangles will be to each other as the base AD of the one is to the base DF of the other. Let the base AD contain two parts of any given measure, and the base DF contain three of the same parts,