so that the contained by two right lines, drawn from the given point to the two points of concourse may be the least possible.

XXXI. To inscribe seven Os in a given O.

XXXII. If the magnitude of a O is given, it is required to find the position of the right line RS, and that of a point P in another line given in position, from whence, if PAB be drawn to cut the circle in A and B, and the perpendiculars AR and BS are drawn to RS, their rectangle will always be of a given magnitude.

XXXIII. Given the four sides, and the difference of the parts into which one of the s is divided by the diagonal; to construct the trapezium, and point out the limits.

XXXIV. Let AB, AC be two right lines given in position, and P a given point; it is required to draw a line through P to meet the given lines in D and E, so that ADXAE may be to DP x PE in a given ratio.

XXXV. An arc of a given being given in magnitude, draw a tangent to it, so that the part of the tangent intercepted between lines drawn from the centre through the extremities of the given arc, may be of a given mag

nitude.

XXXVI. In a given to inscribe a rectangle = to any given right lined figure, not exceeding half the ▲.

XXXVII. To draw a line parallel to a given line, which shall terminate in two other lines given by position, so as to form with them a ▲ equal to a given a rectangle. XXXVIII. To divide a given ▲ into any proposed number of parts, so as to have any given proportion to one another, by means of lines drawn parallel to one of the sides of the A.

XXXIX. To determine the position of any point P, so that lines drawn from thence to the extremities of three right lines AB, CD, EF, given in length and position, shall form three As mutually to one another.

XL. From two given points A, B to draw two lines AB, BC, to meet in a line DE of any kind given by position; so that the sum or difference of their 's shall be to a given

'.

XLI. From two given points A, B to draw two lines AC, BC meeting in a line DE of any kind, given by po

sition; so as to obtain the ratio of two unequal right lines Mm, Nn.

XLII. From two given points A, B to draw two lines AC, BC to meet in a right line DE given by position; so that their difference shall be to a given right line BD.

=

XLIII. Through a given point P, to draw a line GH terminating in a right line AB, and in a line CD of any kind given both by position; so that the rectangle under the parts thereof PG, PH, shall be of a given magnitude.

XLIV. Through a given point P, between two right lines AB, AC given by position, so to draw a line ED, that the sum of the segments (AD+AE) cut off by it from the two former, shall be to a given line RS.

XLV. From the point of intersection P of two given Os D and C, to draw a line PR; that the part thereof QR intercepted by the two peripheries, shall be to a given line AB.

XLVI. Between two right lines NBN, MBM, given by position to apply a line DE of a given length, and which (produced if necessary) shall pass through a given point A equally distant from the said given lines.

XLVII. From a given point P, to the circumference of a given circle C, to draw a right line PBA, so as to be divided in a given ratio by a line RBS of any kind, given in position.

XLVIII. Through a given point P, to draw a line GH, terminating in the circumference of a given O BGA, and in a line MN of any kind given by position; so that the rectangle under the two parts thereof PG, PH, shall be of a given magnitude.

XLIX. From a given point P in the line passing through the centres of two given Os C and O; so to draw a line PD, that the parts thereof DE, FG, intercepted by those circles, shall be in a given ratio, (viz. as p g).

L. To draw a line EC, to make given angles with a line MN, passing through the common centre O of two given Os MEN, KDF, so that the parts thereof CD, ED, intercepted by that line and the two peripheries, shall obtain a given ratio.

LI. To determine a point P, so that three perpendiculars drawn from thence to as many right lines AB,

K

AC, BC, given by position, shall obtain the ratio of three given lines mn and p respectively.

LII. To determine the position of a point P, at which, lines drawn from three given points A, B, C, shall make givens one with another.

LIII. Any two unequal segments AB, CD of a right line AD being given, as well in position as length; to determine a point P in a line MN of any kind given by position; at which the two s APB, CPD, subtended by those segments shall if possible be to each other.

LIII. Two right lines AB, AC being given both in length and position from the point of their concourse A, so to draw another line AI that two perpendiculars IP, IN, falling from the extreme thereof upon the two given lines, shall cut off alternate segments BP, CN, in a given ratio each to the line AI so drawn.

LIV. Between two lines AG, BH, both given in position and length, to draw a line MN, which shall be in a given ratio to each of the segments MG, NH cut off from the two given lines.

LV. To draw a line DF to cut three other lines AB, AG, BC, given by position, so that the two parts thereof DE, EF intercepted by three lines, shall be respectively to two given lines, de, ef.

LVI. Through a given point P, to draw a line DPF, to cut three lines AC, CB, ABb, given by position; so that the parts thereof DE, EF intercepted by those lines shall obtain a given ratio.

LVII. To draw a line ABCD to cut four other lines MONP, NQ, ROT, SMT, given by position; so that the three parts thereof AB, BC, CD, intercepted by those lines, may obtain the ratio of three given lines m, n and p, respectively.

LVIII. To draw two lines CA, CB from a given point C, to terminate in two other lines PM, PN, given in position, and which together with the line AB, joining their extremes, shall form a ▲ ABC similar to a given one, abc.

LVIX. To describe a ▲ PQR equal and similar to a given A p q r, which shall have its angular points placed in three right lines ABC, BD, AE, given by position.

LX. To describe a trapezium similar to a given one efgh having its angular points placed in four right lines, CN, BM, AL, CBAK, given by position.

LXI. In a given to inscribe a A; 1o. such, that the three sides of the A shall be parallel to three lines given in position.

2o. Such, that two sides of the A will be parallel to two lines given in position, and that the third will pass through a given point.

3. Such, that one side of the A will be parallel to a line given in position, and that the other two will pass through two given points.

4. Such, that each side of the triangle shall pass through a given point.

LXII. To inscribe in a given O, a given, a quadrilateral figure,

1. Such, that the four sides shall be par. to right lines given in position.

2o. Such, that three of the sides shall be par. to right lines given in position, and that the fourth shall pass through a given point.

3°. Such, that two of the sides shall be par. to right lines given in position, and that the other two sides shall pass through two given points.

4°. Such, that one side shall be parallel to a right line given in position, and the other three sides shall pass through three given points.

In the following synopsis it has been thought advisable to use abbreviations, as by that means almost all the questions that have been proposed on the solution of triangles are presented to the reader in a very small compass; and the questions are as intelligibly enunciated as if they were at full length.

The three sides of the A are denoted by

The three respctively opposite angles,

The semiperimeter of A

The segments of base b by the perpendicular,

The bisector of B to b,

a, b, c. A, B, C.

L.

The bisector of B to circ. of circumscribing circle,
The radii of inscribed and circumscribing circles, r, R.
The segments of b made by 1,

v, v'.

The bisector of b,

bis. of b.

Segments made by the point of contact of inscr. O, t, t'. Inscribed and circumsbribing circles

In a given ratio,

in. O, cr. O.

Pr.

R'.

Radius of circle touching base and sides produced

I. Given b; B; (S+p) a maximum.

II.

III.

IV.

V.

VI.

r; (a–c) and (A–C).

B; and lines from A, C. cutting a and c
Pr.

2S; B; and bxp, a max.

2S; B and r.

;

B; r; and lines from A and C to centre of in. O.

A, B, C; and the line from B to cut b Pr.

A-C; s-s'; (a+c) (a~c) a max.
atc; r; L.

B; (a–c); and bis. of b.
(a–c); (b-p); and B a max.
b; (a+c); t÷ť.

bp; B; and inscribed

a-c; p-p'; and l.

2 a max.

b; A; and the A divided in a given
rat. by a line par. to c.

b; (a+c); and B divided by bis. of b
into parts whose dif. is a max.
b; A and 2a =c+m, when mis
given line.

VII.

VIII.

IX.

X.

XI.

XII.

XIII.

XIV.

XV.

XVI.

XVII.

XVIII.
XIX.

XX.
XXI.

XXII.

XXIII.

P; as; c-s'.

XXIV.

XXV.

XXVI.

XXVII.

XXVIII.

XXIX.

a+c-b; l; and L-l.

a-s; c- -s' ; b—p.

b; B and a'c a max.

;

p; B, and the rect. of perpendiculars from the end of p on a, c.

bp2

a max; when a given line from B divides the ▲ Pr.

b; ac ; and A.