842 
with respect to quantity, ancfis usually mark- 
ed thus, A : B. 
Of these the first is called the antecedent, 
and the second the consequent. 
84. The measure or quantity of a ratio is 
conceived by considering what part of the 
consequent is the antecedent ; consequently 
it is obtained by dividing the consequent bv 
the antecedent. 
^ 85. T hree magnitudes or quantities, A, B, 
C, are said to be proportional, when the ra- 
tio of the first to the second is the same as 
that of the second to the third. Thus 2, 4, S, 
are proportional ; because 4 is contained in 
8 as many times as 2 is in 4. 
86. Four quantities, A, B, C, D, are said 
to be proportional when the ratio of the first 
A to the second B is the same as the ratio of 
the third C to the fourth D. It is usually 
written A : B : : C : D, or, if expressed in 
numbers, 2 : 4 : : 8 : 16. 
87. Of three proportional quantities, the 
middle one is said to be a mean proportional 
between the other two ; and the last a third 
proportional to the first and second. 
88. Of four proportional quantities, the 
last is said to be a fourth proportional to the 
other three, taken in order. 
89- Ratio of equality is that which equal 
numbers bear to each other. 
90. Inverse ratio is when the antecedent is 
made the consequent, and the consequent the 
antecedent. Thus, if 1 ; 2 ; ; 3 l 6 ; then in- 
versely, 2 * 1 * • s * 3. 
. Alternate, proportion is when antecedent 
is compared with antecedent, and consequent 
with consequent. Thus, if 2 * 1 * * 6 * 3 • then 
by alternation 2 * 6 ; 1 ; 3. * 
92. Proportion by composition is when the 
antecedent and consequent, taken as one quan- 
tity’ are compared either with the consequent 
or with the antecedent. Thus, if 2 [ 1 '• 6 ’ 3; 
then by composition 2 4- 1 ; 1 • •* 6 _L* 3 '* 3 : 
and 2-q-l *2*6-j-3*6. * " ' " 
93. Divided proportion is when the difference 
of the antecedent and consequent is compared 
either with the consequent or with the antece- 
dent. Thus, if 3 : 1 ;; 12 : 4 ; then, by divi- 
sion, 3 — 1 : 1 ;; 12 - 4 : 4, and 3 — 1 * 3 
*; 12 — 4 ; 12. 
94. Continued proportion is when the first is 
to. the second as the second to the third; as the 
third to the fourth ; as the fourth to the fifth ; 
and so on. 
95. Compound ratio is formed by the multi- 
plication of several antecedents and the several 
consequents of ratios together, in the following 
manner : 
If A be to B as 3 to 5, B to C as 5 to 8, 
and C to D as 8 to 6; then A will be D as 
3X5X8 120 _ , . 
= 94o = » 5 that 1S > A .* D ; ; i ; 2. 
GEOMETRY. 
101. Scales of equal parts. A scale of 
equal parts is only a straight line, divided 
into any number of equal parts at pleasure. 
Each part may represent any measure you 
please, as an inch, a foot, a yard, &c. One 
of these is generally subdivided into parts of 
the next denomination, or into tenths or hun- 
dredths. Scales may be constructed in a 
variety of ways, 'i lie most usual manner is 
to make an inch, or some aliquot part of an 
inch, to represent a foot ; and then they are 
called inch scales, three-quarter-inch scales, 
half-inch scales, quarter-inch scales, &c. 
They are usually drawn upon ivory or box- 
wood. See Instruments. 
102. An ax ion is a manifest truth, not re- 
quiring any demonstration. 
103. Postulates are things required to be 
granted true, before we proceed to demon- 
strate a proposition. 
. 104. A proposition is when something is 
either proposed to be done, or to he demon- 
strated, and is either a problem or a theo- 
rem. 
105. A problem is when something is pro- 
posed to be done, as some figure to be 
drawn, 
106. A theorem is when something is pro- 
posed to be demonstrated or proved. 
107. A lemma is when a premise is de- 
pnonstrated, in order to render the thing in 
hand the more easy. 
108. A corollary is an inference drawn 
from the demonstration of some proposition. 
109. A scholium is when some remark or 
observation is made upon something men- 
tioned before. 
110. The sign = denotes that the quanti- 
ties betwixt which it stands are equal. 
111. The sign -f denotes that the quantity 
after it, is to be added to that immediately 
before it. 
1 12. T lie sign — denotes that the quantity 
after it is to be taken away, or subtracted 
from, the quantity preceding it. 
From the point f, with any radius, describe 
the arc d e, cutting AB in e and d. From the 
points ed, with the same or any other radius 
describe two arcs, cutting each other in g. 
Through the points f and g, draw the line f g, 
and f C will be the perpendicular required. 
Prob. 4. To make an angle equal to another 
angle which is given, as a Bb. 
1 10 m the point B, with any radius, describe? 
the arc ab, cutting the legs B a, Bb, in the 
points a and b. Draw the line I)e, and from 
the point D, with the same radius as before, 
describe the arc ef, cutting Deine. Take 
the distance b a, and apply it to the arc ef 
trom e to t. Lastly, through the points D f 
draw the line D f, and the angle e Df will be 
equal to the angle bBa, as was required. 
Prob. 5. To divide a given angle, ABC 
into two equal angles. 
from the pomt B, with any radius, describe, 
the aic AC. f rom A and C with the same, or 
any other radius, describe arcs cutting each, 
m d. Draw the line B d, and it will bisect 
e angle ABC, as was required. 
Prob. 6. To lay down an a: 
GEOMETracAL Problems. 
5 X 8 X 6 240 
96. Bisect means to divide any thing into 
two equal parts. 
97. Trisect is to divide any thing into 
three equal parts. 
98. Inscribe, to draw one figure within an- 
other, so that all angles of the inner figure 
touch either the angles, sides, or planes, of the 
external figure. 
99. Circumscribe, to draw a figure round 
another, so that either the angles, sides, or 
planes of the circumscribed figure, touch all 
the angles of the figure within it. 
100. Rectangle under any two lines, means 
a rectangle which has two of its sides equal 
to one of the lines, and two of them equal to 
the other. Also the rectangle under AB, 
CD, means AB x CD. 
Prob. 1. To divide a given line AB into 
two equal parts. 
From the points A and B as centres, and 
with any opening of the compasses greater 
than half AB, describe arches, cutting each 
other in c and d. Draw the line cd; and 
the point E, where it cuts AB, will be the 
middle required. 
Prob. 2. 'I o raise a perpendicular to a 
given line AB, from a point given at C. 
Case 1. When the given point is near the 
middle of the line on each side of the point 
C. Take any two equal distances, Cdand 
C e; from d, and e, with any radius or open- 
ing ot the compasses greater than Cd, or 
C e, describe two arcs cutting each other in 
f. Lastly, through the points f, C, draw the 
line f C, and it will be the perpendicular re- 
quired. 
Case 2. When the point is at, or near, the 
end of the line. Take any point d, above 
the line, and with the radius or distance d C, 
describe the arc eC f, cutting AB in c and C. 
I hrough the centre d, and the point e, draw 
the line edf, cutting the arc eCfinf. 
Through the points f, 0, draw the line f C, 
and it will be the perpendicular required. 
Prob. 3. From a given point f, to let fall 
a perpendicular upon a given line AB. 
T down an angle of any, 
number of degrees. J 
There are various methods of dorno- this.. 
One is by the use of an instrument called a 
protractor, with a semicircle of brass havino- 
its circumference divided into degrees. L t 
AB be a given line, and let it be required U> 
draw from the angular point A, a line making 
w‘th AB any number of degrees, suppose 20. 
Day the straight side of the protractor alon°- 
the line AB, and count 20° from the end if 
of the semicircle; at C, which is 20 9 from B 
mark; then, removing the protractor, draw 
the line AC, which makes with AB the angle 
required. Or, it may be done by a divided- 
line, usually drawn upon scales, called a line 
ot chords. Take 60' 1 from the line of chords 
m the compasses, and setting one at the an 
gujar point B, prob. 4, with that opening ns a 
radius, describe an arch, asab: then take 
the number of degrees you intend the angle 
to be of, and set it from b to a, then isaB b 
the angle required. See Instruments. 
Prob. 7. Through a given point C, to draw 
a line parallel to a given line AB. 
Case 1. Take any point d, in AB; upon 
d and C, with the distance Cd, describe t'\c>. 
arcs eC, and df, cutting the line AB in c 
andd Makedf equal to eC; through C 
and f draw C f, and it will be the line re- 
quired. 
. ^ ase 2. When the parallel is to be at a 
given distance from AB. From any two 
points, c and d, in the line AB, with a radius 
equal to the given distance, describe the arcs 
e and f : draw the line CB to touch those 
arcs without cutting them, and it will be pa- 
rallel to AB, as was required. 
Prob. 8. To divide a given line AB, into 
any proposed number of equal parts. 
From A, one end of the line, draw A c 
making any angle with AB ; and from B the 
other end, draw Bd, making the amde ABd 
equal to BAc. In each ot these lines, Ac 
Bd, beginning at A and B, setoff as many 
equal parts of any length as AB is to be di- 
!u°‘ , Joi , n , the P oints C 5 > 57, and 
AB will be divided as required. 
Prob. 9. To find the centre of a given cir- 
cle, or ot any one already described. Draw 
any chord AB, and bisect it with the perpen- 
dicular CD. Bisect CD with the diameter 
5 
