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(© Triangles which stand upon the same ters, the isosceles has the greatest area

or upon equal bases, and between the
same parallels, are equal to one ano. (D) Problems relating to the Triangle.

17 (a) To describe a triangle, when there
(d) Triangles which have equal altitudes are giren-
are to one another as their bases, and

1. Two sides and the included angle.
triangles which have equal bases, as

2. Two sides, and an angle opposite
their altitudes; also any two triangles

to one of them.
are to one another in the ratio, which

3. Two angles and the interjacent
is compounded of the ratios of their

bases and altitudes

65, 66

4. Two angles and a side opposite
(e) Triangles which have one angle of

to one of them.
the one equal to one angle of the

5. The three sides

other are to another in the ratio, which (6) To describe a triangle, when there
is compounded of the ratios of the

are given the
sides about the equal angles, or as the

1. Vertical angle, base, and sum (or.
rectangles under those sides 66

difference) of the two sides.
( Triangles which have one angle of

2. Vertical angle, base, and area.
the one equal to one angle of the other,

3. Vertical angle, sum (or differ-
and the sides about the equal angles

ence) of sides and area.
reciprocally proportional, are equal to

4. Base, sum (or difference) of sides
one another; and, conversely, equal

and area

triangles, which have one angle of the (c) To describe a triangle-
one equal to one angle of the other,

1. Which shall be equal to a given
have the sides about the equal angles

rectilineal figure, and have a side
reciprocally proportional : 66

and adjoining angle the same
(9) Two triangles are similar, when

with a given side, and adjoining
angle of the figure

1. The three angles of the one

2. Which shall be equiangular with
equal to the three angles of the

a given triangle, and have a given
other, each to each

perimeter (or area]

or 2. The three sides of the one pro-

3. Which shall have for two of its
portional to the three sides of the

sides the parts which are cut off


by the third side from two straight
or 3. One angle of the one equal to

lines given in position, and the
one angle of the other, and the

third passing through a given
sides about the equal angles pro-



60 (d) To describe a right-angled triangle
or 4. One angle of the one equal to which shall have its three sides pro-
one angle of the other, and the portionals

sides about two other angles

See « Circle.":
proportionals, and the remaining
angles of the same affection, or

Triplicate, one ratio said to be of another 34
one of them a right angle 61

Trisection of an arc or angle, under what
(h) Similar triangles are to one another

form the problem has been put 113
in the duplicate ratio (or as the
squares) of their homologous sides 67

Undecagon. See “ Hendecagon."
(1) Of all triangl having the same

Ungula. See “ Sphere.”
two sides, that one has the greatest

Unit of length, or linear unit, is any arbitrary
area, in which the angle contained by

straight line, as an inch, a foot.
the two sides is a right angle 103

-of surface, is the square of the linear

(k) Of triangles which have equal bases,

sch. 18
and equal areas, the isosceles has the

--of content, is the cube of the linear unit
least perimeter; and of triangles

sch. 142
having equal bases and equal perime- Variation, a short form of expressing certain


sch. 62
Now the area of the triangle A B C is equal to

Vertex of a triangle 2, polyhedron 126,
LF X A K, for it is equal to half the rectangle pyramid 127, cone 267.
under the radius L F of the inscribed circle, and

of the diameter of a conic section 220
the sum 2 A K of the three sides (I. 26. cor.); also
BF x B K is equal to LF X MK, because, L BM

-a term in perspective projection 208
being a right angle, the right-angled triangles BFL, -of a conic section

M K B are similar: but AK XAF : AKXFL Vertical plane, in perspective projection 208
:: AF: FL, i.e. :: AK: KM, i.e. : : AKX

Vertical anglé, of a triangle. See“ Triangle.”
FL: KMX FL : therefore A KX F L is a mean

proportional between A K X AF and KM * F L or

See" Content."
B F BK, that is (if a, b, c represent the three
sides opposite to the angles A, B, C respectively and

Wedge, spherical. See “ Sphere."
S the half of (a+b+c) the area of the triangle is a
mean proportional between Sx (S –a) and (s - b)
X (S-c).

Zone, spherical. See “ Sphere."


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The term Trigonometry is derived from the two Greek words, Tplywvov, a triangle, and uerpéw, I measure, and originally signified simply the science by which those relations are determined which the sides and angles of a triangle bear to each other, being called plane or spherical trigonometry according as the triangle was described on a plane or spherical surface. At present, however, the term has a much more extensive signification, as the science now embraces all the theorems expressing the relations between angles and those functions of them to be hereinafter described ; the terms plane and spherical still denoting those branches of the science immediately connected respectively with plane and spherical triangles. The present treatise will be found to contain the fundamental theorems of the science, with their applications to plane trigonometry and to the construction of trigonometrical tables. A knowledge of these theorems should be acquired by the student before he proceeds to the Differential Calculus; the remainder of the subject, as a branch (and a most important one) of pure analysis, he may, in many respects, read more advantageously when some knowledge of the Calculus shall have prepared him to enter on a wider field of analytical investigation.

The ancients, it is well known, cultivated astronomy with considerable assiduity and success; but as little advance could be made in it without a knowledge of trigonometry, their cultivation of this branch of mathematics was necessarily co-existent with that of astronomical science. Little, however, of what was written by them on this subject is come down to us.

During the dark ages which overshadowed the nations of Europe, trigonometry, in common with other sciences, seems to have made some little progress among the Arabians, from whom it was derived by the Europeans after the revival of literature and science among them, about the beginning of the fifteenth century. After this period, the attention of scientific men, in imitation of the ancients, was principally directed to astronomy; and those to whom that science was most indebted for its progress were those to whom trigonometry also was equally indebted. Among the first of these may be mentioned George Purbach, professor of mathematics and astronomy at the University of Vienna, and John Müller, his pupil and successor, sometimes called Regiomontanus, from Mons Regius, or Koningsberg, a small town in Franconia, the place of his nativity. The former was born in 1423, and died

in 1462 ; the latter was born in 1436, and died in 1476. Copernicus also, the celebrated astronomer, wrote a treatise on trigonometry about the year 1500. Several others might also be mentioned, the greater part of whom were natives of Germany, the progress of astronomy and trigonometry, for a considerable period after their revival in Europe, having been due very principally to the philosophers of that country. Vieta, a native of France, was born in 1540, and was one of the first mathematicians of his time. He gave improved methods of calculating trigonometrical tables, and enriched the science with a variety of theorems. He appears to have been the first who made any considerable application of algebra to this subject.

The following authors also wrote on trigonometry: George Joachim Rheticus, professor of mathematics in the University of Wittemburg, who died in 1576; Pitiscus; Valentine Otho, mathematician to the electoral Prince Palatine; and Christopher Clavius, a German Jesuit. These authors lived during the latter part of the sixteenth century.

At the period we have just mentioned, all the fundamental formulæ of trigonometry, and their applications to the calculation of tables and the sides and angles of triangles, were well known; but the immense progress of modern analysis has since opened a wide field for the applications of trigonometry, beyond those primary objects of the science to which the first part of this treatise will be devoted. Further historical notices will be reserved, to be made in immediate connexion with such parts of the subject as they may tend to elucidate, or render more interesting to the student.

SECTION I. Trigonometrical definition of an Angle-Complements and Supplements

of Angles, and of the arcs subtending themNumerical measure of Angular space-Sexagesimal and Centesimal divisions of the Circle.

(1.) Def. An angle, in geometry, denotes the inclination of one straight line to another, and, in this simple acceptation, must be less than two right angles. In trigonometry, the term has a more extended meaning. (See Geom. III. $ 2. Prop. 13. Schol.)

Let Č A be a fixed line, and C a given point in it; and suppose CP to revolve in one plane about C, coinciding at first with C A ; then is the whole angular space described by CP in its revolution about C, called an angle, which may, therefore, in this case, be of any magnitude whatever; also, if with the centre C and any radius we describe a circular arc, sub

A tending any angle A CP, this arc cannot, according to the geometrical definition of an angle, be greater than the semi-circumference of the circle; but, according to the trigonometrical definition of an angle, the subtending arc may be of any magnitude, consisting of any number of circumferences, or any part of a circumference.

(2.) If we denote the angle ACP by A, the subtending arc by a, the sum of two right angles by Ty, and the length of the semicircumference of the circle by II,

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