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Cales. Give!). Sought.

Solution.

Spo. Tris. B, C and AB, BC the fideR: cos. B :: tan. AB:tan. BD,

two angles between the (cale 2.), and tan. C:tan. B:: 6

and the side angles. fin. BD : fin. DC; and DC is
opposite to

less than DB, if B and C be
one of them

of diff. affection; or leís than
C.

the supplernent of DB, if B
and C be of the same aff-c-
tion. In other cases, DC is
ambiguous. If B and C be
of the same affection, BC is
the sum of BD, DC; other-

wise it is their difference.
AB, BC, and one of R: cos. B :: tan. AB : tan. BD

B, two sides the other (case 2.), and the difference of 7 and the angles.

BC and BD is DC. Andi included

sin. DC : fin. DB : tan. B :
angle.

tan. G, (p. 3.), and B, C are
of the fame affection, if BC
be greater than BD; other-

wise they are of diff. affection.
AB, BC, and AC the third Find BD and DC as in the last
B, two sides side.

case, then cos. BD:cos. DC:: 8 and the in

cos. BA : cos. AC (p. 3). In cluded

BD, DC be of the same af-
angle.

fection, BA, AC are of the
fame affection

;

otherwisel

they are of different affection. A,B, and AB, C the third R:cos. AB::tan. B:cot. BAD,

two angles angle. (case 3.), and the diff. of BAC, 9 and the in

BAD is DAC, then sin.
cluded fide.

BAD : fin. DAC:: cos. B:
cos. C, (p. 3.), if BAC bel
greater than BAD, B, C are
of the same affection; other-
wise they are of different af-

fection.
JA, B, and AC one of Find BAD and DAC, as in the
AB, two the other last case; then cos.

DAC :
angles and fides. cos. BAD: : tan. AB: tan.
the inclu-

AC. If DAC, B be of the
ded fide,

same affection, AC is less than
90°; otherwise it is greater

than 90°.
Mm

Cases,

10

Spb. Trig. Cafes.

II

:

Given.
Sought.

Solution.
AB, AC, BCB one of the Let the perp. AD fall within,
the three angles.

or be the nearest to B or C fides.

that falls without, tan. BC: tan. Ź sum of BA, AC :: tan. diff. of BA, AC : tan. | E, and IE added to BC, gives the segment nearest the greater fide, if the sum of AB, AC be less than 180°; otherwise it gives the seg. ment nearest the less side. (Prop. 22.). And tan. AB : tan. BD : Ri

cos. B. (case 12.) Otherwise, Let D be the diff.

of AB, BC; then the rect. fin. AB, fin. BC : rect. fin. sum and diff. of D, and

AC:: R2:: fin 2. B.(P.23). Otherwise, Let P be the pere.

meter; then rect. fin. AB, fin. BC: rect. fin. P. fin. diff. of

P, AC:: R2:cosa, B.(24). A, B and CAC one of With the supplement of either the three the sides. of the angles A, C, and the

measures of the other two angles, suppose a triangle made ; and in it find the angle opposite to the side which is the measure of the angle at B, and the measure of the angle thus found is AC.

12 angles.

NOTES

N O T E S.

BOOK. !.

DEFINITIONS.

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HE ist definition wants a condition to make it complete, Book I.

for to have no magnitude is not peculiar to a point : This condition is now inserted from Dr Hooke, who says, that a point has position, and a relation to magnitude, but has itself no magnitude: It may also be said to be an indivisible mark in magnitude, as Tacquet has it: Or, it may be said to be a fign ufed for determining position and the extremities of lines, for the name onestoy appears to have been given to it from its use.

The 8th definition is left out; because it does not belong to the Elements ; nor can it be explained, so as to be understood by beginners, as is observed by Dr Simson.

The 13th definition is also omitted, because it is useless in a translation, its only design being to explain a Greek word.

And the 19th, which is the definition of a segment, is left out here, because it is given in the third book, which is its proper place.

And the definition of the radius of a circle is introduced, be cause it is very frequently used by Geometers.

These are all the alterations that have been made in the defi. nitions of this book ; but many more might have been made with propriety. The first nine definițions might have beer given in the form of an introduction, for they are none of them geometrical, except the seventh, as amended by Dr Simson. The terms by which a line and a superficies are defined, give fome explanation of the meaning of these words, but give no geometrical criteria by which to know them; and the best way of acquiring proper ideas of them, is by considering their relation to a solid, and to one another, as Dr Simson has done.

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BOOK I. “It is neceffary to consider a solid, that is, a magnitude which

has length, breadth, and thickness, in order to understand a

right the definitions of a point, line, and superficies; for all these arise from a solid, and exist in it: The boundary, or boundaries which contain a solid, are called superficies, or the boundary which is common to two solids, which are contiguous, or which divides one folid into two contiguous parts, is called a fuperficies: Thus, if BCGF, be one of the boundaries which contain the solid ABCDEFGH, or which is the common boundary of this solid, and the solid BKLCFNMG, and is therefore in the one as well as the other folid, it is called a superficies, and has no thickness : For if it has any, this thickness inuft either

of the thickness of the solid AG, or the folid BM, or a part of the thickness of each of them. It cannot be a part of the thickness of the folid BM; becaute, if this folid be removed from the solid AG, the superficies BCCF, the boundary of the folid AG remains still the same as it was. Nor can it be a part of the thickness of the solid AG; because, if this be removed from the solid BM, the superficies BCGF, the boundary of the solid BM, does nevertheless remain ; therefore the superficies BCGF has no thickness, but only length and breadth.

The boundary of a superficies is called a line; or a line is the common boun

I dary of two fuperficies that are conti 71

N guous; or it is that which divides one fuperficies into two contiguous parts : Thus, if BC be one of the boundaries which contain the superficies ABCD, A B or which is the common boundary of this fuperficies, and of the superfícies KBCL, which is conti guous to it, this boundary BC is called a line, and has no breadth : For, if it has any, this must be part eicher of the breadth of the superficies ABCD, or of the superficies KBCL, or part of each of them. . It is not part of the breadth of the fuperficies KBCL ; foc, if this fuperficies be removed from the Superficies ABCD, the line BC, which is the boundary of the superficies ABCD, remains the same as it was : Nor can the breadth that BC is supposed to have be a part of the breadth of the superficies ABCD, because, if this be removed from the superficies KBCL, the line BC, which is the boundary of the fuperficies KBCL, does nevertheless remain ; therefore the line BC has no breadth: and because the line BC is in a superficies, and that a superficies has no thickness, as was thewn; therefore a line has neišlier breadth nor thickness, but only length.

The boundary of a line is called a point, or a point is the common boundary or extremity of two lines that are contiguous :

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Thus, if B be the extremity of the line AB, or the common Book I.
extremity of the two lines AB, BK, this extremity is called a
point, and has no length : For, if it has any, this length must
either be part of the length of the line AB, or of the line BK.
It is not part of the length of the line KB ; for, if the line KB
be removed from AB, the point B, which is the extremity of the
line AB, remains the same as it was : Nor is it part of the
length of the line AB; for, if AB be removed from the line
KB, the point B, which is the extremity of the line KB, does
nevertheless remain; therefore the point B has no length: and
because a point is in a line, and a line has neither breadth nor
thickness; therefore a point has no length, breadth, nor thickness.
And in this manner, the definitions of a point, line, and super-
ficies, are to be understood.”

No definition of a straight line has been given that is unexcep-
tionable, though many of the ancients attempted it, as Proctus
observes, who has allo preserved their definitions. That given
in the Elements, viz. its lying evenly, equally, or uniformly
between its extremities, expresses the nature of a straight line
too metaphysically: its meaning is, that a Itraight line has not a
convex and a concave fide; but that both sides are alike. But the
distinguishing character of a straight line, according to Euclid, is,
that it is impossible to apply one part of it to another, or one
straight line to another, without their coinciding. All other
lines require some artifice in applying them to one another, its
order to make them coincide ; but no such artifice is necessary in
the case of straight lines, for they always coincide in whatever
way we proceed to apply them to one another. That this was
Euclid's idea of a straight line, is manifest from the fourth and
eighth propofitions of the first book, in which he does not few
how the sides of tħe triangles are to be applied to one another, so
as to coincide, but takes it for granted, that AB in the fourth
shall lie along DE, and that BC in the eighth fhall lie along EF,
as soon as they are applied to one another. Now, this facility of
application follows immediately from the uniformity of the sides
of a straight line, and the other properties of a straight line are
easily deduced from it.

Plato's definition, that the extremity of a straight line casts a
shadow along the whole line; and that of Archimedes, that a
straight line is the least of all the lines which have the same ex-
tremities with it, were evidently designed for particular purposes,
and are not fit for the Elements.
The other definitions mentioned by Proctus, may be reduced

One of them is, that a straight line is that of which
the position is determined by the position of any two points of it.
This is a property of straight lines which is supposed in the first

and

to two.

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