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these four parts are equal to one another (by Def. 11.) and so EB a Quadrant, or fourth part of the Cir. cumference ; therefore the Radius EC is always the Sine of the Quadrant, or fourth part of the Circle EB.
Sines are said to be of so many Degrees, as the Arch contains parts of the 360, into which the Circumference is supposed to be divided; so the Radius being the Sine of a Quadrant, or fourth part of the Circumference, which contains 90 De- . grees; (the fourth part of 360) therefore the Radius must be the Sine of 90 Degrees.
22. That part of the Radius comprehended between the Extremity of the right Şine and the lower End of the Arch, viz. DB, is called the versed Sine of the Arch A B.
23. If to any Point in the Circumference, viz. B, ' there be drawn a Diameter FCB, and from the point B perpendicular to that Diameter, there be drawn the Line BH; that Line is called a Tangent to the Circle in the point B; which Tangent can touch the Circle only in one point B, else if it touch'd it in more, it would go within it, and so not be a Tangent but a Chord (by Art. 18.)
24. The Tangent of any Arch AB, is a right Line drawn perpendicular to a Diameter through the one end of the Arch B, and terminated by a Line CAH, drawn from the Center through the other end A ; thus BH is the Tangent of the Arch AB.
25. And the Line which terminates the Tangent, viz, CH, is called the Secant of the Arch AB.
26. What an Arch wants of a Quadrant is called the Complement of that Arch; thus A E being what the Arch AB wants of the Quadrant EB ; is called the Complement of the Arch A B.
27. And what an Arch wants of a Semicircle is called the Supplement of that Arch ; thus since AF
is what the Arch AB wants of the Semicircle BAF, it is called the Supplement of the Arch A B.
28. The Sine, Tangent, &c. of the Complement of any Arch, is called the Co-Sine, Co-Tangent, &c. of that Arch; thus the Sine, Tangent, &c. of the Arch AE is called the Co-Sine, Co-Tangent, &c. of the Arch AB.
29. The Sine of the Supplement of an Arch is the fame with the Sine of the Arch itself, for drawing them according to the Definitions, there results the self same Line.
30. A right lin’d Angle is measured by an Arch of a Circle described upon the angular Point as
D a Center, comprehend- A ed between the two Legs
E that form the Angle ; thus the Angle ABD is measured by the Arch
B AD of the Circle CADE that is described upon the point B as a Center ; and the Angle is said to be of as many Degrees as the Arch is ; so if the Arch A D be 45 Degrees, then the Angle ABD is said to be an Angle of 45 Degrees.
Hence Angles are greater or less according as the Arch described about the angular Point, and terminated by the two Legs, contain a greater or less Number of Degrees.
31. When one Line falls perpendicularly on another, (as A B on CD)
A then the Angles are right; (by the 11th) and defcri. bing a Circle on the Center B, since the Angles с B D AB C, A B D are equal, their measures must be so too, i. e. the Arches AC, AD must be equal; but the whole CAD is a Semicircle, since CD, a Line passing through the Center B, is a Diameter, therefore each of the
parts AC, A D is a Quadrant, i.e. 90 Degrees ; fo the measure of a right Angle is always 90 Degrees. 32. If one Line AB fall any way upon another,
CD, then the Sum of the two Angles ABC,
ABD is always equal B D
to the Sum of two right
Angles. For on the point B, describing the Circle CAD, it is plain, that CAD is a Semicircle (by '15th); but CAD is equal to CA and A D the measures of the two Angles; therefore the Sum of the two Angles is equal to a Semicircle, that is, to two right Angles (by the last).
Cor. 1. From whence it is plain, that all the Angles which can be made from a point in any Line, towards one side of the Line; are equal to two right Angles.
2. And that all the Angles which can be made about a Point, are equal to four right ones. 33. If one Line AC cross another B D in the
Point E, then the opposite Angles are equal, viz. BEA to CED, and
BEC equal to A ED. For upon B
the point E, as a Center, describ-
ABC is a Semicircle, as also BCD (by 15th) therefore the Arch ABC is equal to the Arch BCD; and from both taking the common Arch BC, there will remain A B equal to CD, 1. e. the Angle BEA equal to the Angle CED (by Art. 30.). After the same manner we may prove, that the Angle BEC is equal to the Angle AED.
34. Lines which are
A equally diftant from
D 35. If a Line GH cross two Parallels A B, CD, then the external Angles are equal, viz. GEB equal to CFH and AEG equal to HFD. For since A B and CD are parallel to one another, they may be considered as one broad Line, and GH crossing it; then the vertical or opposite Angles GEB, CFH are equal (by the 33d) as also AĚG and HFD by the same.
36. If a Line GH cross two Parallels A B, CD then the alternate An
G gles, viz. A EF and EFD, or CFE and
E FEB are equal; that A
-В is, the Angle A EF is equal to the Angle C
D EFD, and the An
F gle CFE is equal to the Angle FEB, for H Н GEB is equal to AEF (by the 33d.) and CFH is equal to E F D by the same, but GE B is equal to CFH by the last. Therefore AEF is equal to EFD; the same way we may prove FEB equal to EFC.
37. If a Line G H cross two parallel Lines A B, CD, then the external Angle GE B is equal to the internal opposite one EFD, or GEA equal to CFE. For the Angle A EF is equal to the Angle EFD by the last ; but AEF is equal GE B (by the 33d) therefore GEB is equal to EFD; the faine way we may prove AEG equal to CFE.
38. If a Line GH cross two parallel Lines AB, CD, then the Sum of the two internal Angles, viz. BEF and DFE, or A EF and CFE are equal to two right Angles; for since the Angle GEB is equal to the Angle EFD (by the last) to both add the Angle FEB, then GEB and BEF_are equal to BEF and DFE ; but GE B and BEF are equal to two right Angles (by the 32d) therefore BEF and D FĚ are also equal to two right Angles. The same way we may prove that AEF and CFE are equal in two right Angles.
39. A Figure is any part of Space bounded by Lines or a Line. If the bounding Lines be streight, it is called a Rectilineal Figure as A; if they be curved, it is called a curvilineal Figure as B or C; if they be partly curve Lines and partly streight, it is called a mixt Figure as D.
40. The most simple recti
lineal Figure is that which is A bounded by three right Lines,
and is called a Triangle, as A. 41. Triangles are divided into different kinds, both with respect to their Sides and Angles : with respect to their Sides they are commonly divided into three kinds, viz.
42. A Triangle having all it's three Sides equal to one another, is called an Equilateral Triangle, as A.
43. A Triangle having two of it's Sides equal to one another, and the third Side not equal to either of them, is called an Isosceles Triangle, as B.
44. A Triangle having none of it’s Sides equal to one another, is called a Scalene Triangle, as C.