The Formulæ of Plane Trigonometry

Preface

This Pamphlet is published with two objects: first, to exhibit a specimen of a collection of “Forumulæ of Pure Mathematics,” which I am preparing for publication; secondly, to suggest the substitution of symbols for the cumbrous expressions “sin,” “tan,” “cosec,” &c., at present employed in Trigonometry.

As to the first of these objects, I am in hopes that, by publishing this specimen by itself, I may receive, from other mathematicians, suggestions both as to the form and the matter of the larger work; and any simple and compendious formulæ, not given in the published treatises, or superseding those which are there given, will be most acceptable for this object.

As to the second, I am anxious to ascertain what probability there is of others’ consenting to adopt these, or any similar symbols, before employing them either in the above-mentioned “Formulæ of Pure Mathematics,” or in another work, (also in preparation,) on Algebraical Geometry, (on the plan indicated in a Syllabus published in 1860,) into both of which works these symbols would of course enter largely.

Objections may be raised, first, against the introduction of any symbols at all into this subject, and, secondly, against the particular symbols here suggested: these objections I proceed to examine.

First, then, as to the introduction of any symbols at all: if it could be shown that such symbols were likely to confuse those who are used to the existing notation, or to make the books at present in use unintelligible to those who learn only the new notation, I grant that it would be much too late in the history of Trigonometry to attempt any such radical change. But I am sure that no such effects would be produced; the new symbols would still be called “sin,” “cos,” &c., so that the old names would not die out, and those who use the new symbols would find no more difficulty in reading books where the names are given in full, than we do in reading old mathematical books where the words “plus,” and “minus” occur. And there can be no question that the use of symbols would save much time, space, and labour; all of which are points of great importance to those who are much engaged in writing or printing Mathematics.

Secondly, as to the particular symbols here suggested: on this point I speak much more doubtfully; and I shall be very willing, if simpler and more appropriate symbols be proposed, to adopt them instead of these; still, as these have not been chosen without some thought, and the trial and rejection of many others, I have great hopes that they may be found sufficiently simple and expressive for the purpose: and perhaps a simple account of the way in which they originated will be the best explanation and defence of them which I can offer.

It seemed necessary, then, that symbols for such a purpose should be—easily written—suggestive (as far as possible) of their meaning—connected with each other—and, above all, distinct from all symbols at present in use. Letters of the alphabet of course suggested themselves first: but besides that all the alphabets, capital and small, Roman, Greek, even Old English and Hebrew, are already largely appropriated in Mathematics, it did not seem that any but initials were sufficiently connected with the words to stand much chance of being remembered. Now the words “sine, secant, cosine, cosecant, cotangent, tangent,” have unfortunately only three initials among them; and of these “S” and “C” (whether we take them as capital or small letters,) are already distinctly appropriated in this very subject of Algebraical Geometry—“S” and “C” signifying, one, the semi-perimeter of a triangle; the other, one of its angles—and “s” and “c” signifying, one, an arc of a curve; the other, one of the sides of a triangle: as to the latter indeed it is sufficient to point out that the phrase “ $a . cos A + b . cos B + c . cos C$ ” would, under this notation, be written “ $a . c A + b . c B + c . c C$ ”!

Setting aside, then, all alphabets, and (of course) all purely mathematical symbols, such as numerals, &c., no course remained but to invent new symbols for this purpose. In setting about this I took as principles—to secure their being easily written, that each should consist of two strokes of the pen only—to connect them with each other, that one of these two strokes should be the same in all—to make them suggestive of their meaning, that they should represent (as nearly as possible) the geometrical lines to which these ratios belong. And as, in modern Trigonometry, each of these ratios involves two lines, I thought it better to fall back on the older theory, where the sine, cosine, &c., are all single lines.

A semicircle with centre O. The diameter is at the bottom, its right end point is A. The point B is on the circle. N is on the diameter perpendicular below B, T is the intersection of OB and the perpendicular in A.

In the annexed diagram, $B N$ is the sine of the angle $A O B$ , $O N$ its cosine, $T O$ its secant, and $A T$ its tangent. It occurred to me, then, to construct a set of symbols for these by taking the semicircle itself as the part common to all the symbols, and these lines as their distinctive features, altering them only so far as to make each symbol symmetrical. I shall now consider each “goniometrical ratio” by itself.

I. “Sine,” . In this it will be seen that the line $B N$ has merely been shifted to the middle of the figure.

II. “Cosine,” . In this the line $O N$ has been produced, (which in hasty writing one could hardly avoid doing,) and taken a little beyond the curve, to avoid confusion with the existing symbol for “semicircle.” Observe here also that the line in this symbol is at right angles to that in the last; which may be connected with the fact that these are corresponding ratios of complemental angles (i. e. of angles, whose sum is a right angle).

III. “Secant,” . In this the line $T O$ has been produced downwards, to avoid all chance of confusion with the symbol .

IV. “Cosecant,” . This I derived from the former, by turning the line round through a right angle, on the principle which I have explained in II.

V. “Tangent,” . If the symbol for this were taken from the diagram, it would have a one-sided effect, and it would be difficult to find an analogous symbol for “cotangent,” since the three other positions in which it might be placed are pre-occupied by the letters h, μ, and y. I therefore preferred placing it horizontally on the top of the curve, leaving a little interval between the two, to avoid confusion with the letter π.

VI. “Cotangent,” . I could not derive this from the last, on the principle of II, without destroying its character as a tangent. I therefore simply inverted the former symbol, which may be taken to indicate the fact that each is the reciprocal of the other: so that if $= \frac{a}{b}$ , $= \frac{b}{a}$ , which seems to be a very consistent and self-interpreting notation.

VII. “Versed-sine,” . This is of course a combination of the semicircle with the letter “V”; it contains one stroke more than the other symbols, but as it is very seldom used, this is of little importance.

I have thus endeavoured to show that the proposed symbols satisfy three of the four requisitions, viz.: that they should be—easily written—suggestive of their meaning—and connected with each other. As to whether they are distinct from existing symbols, it is for the objectors to point out any with which they are liable to be confused; none such have been suggested to me, except the letter $Ω$ , which is something like the new symbol ; but, as the latter is closed, instead of open, below, and much wider in proportion to its height than the former, I do not think there is much danger of either’s being mistaken for the other.

I will conclude by putting into the form of definite questions the points to which I wish to draw the attention of those mathematicians into whose hands this may come:

(1) Do you object in limine to the introduction of any symbols whatever as substitutes for the words sin, cos, &c.? If not,

(2) Can you suggest others, better adapted than these for such a purpose? If not,

(3) Do you so far approve of the symbols here suggested, that, if they were employed in a published work, you would not, on that account alone, object to use or to recommend such a work?

Ch. Ch. June 11, 1861.

Preliminary Remarks.

The subject-matter usually assigned to Plane Trigonometry may be more properly arranged thus:

Part I. Goniometry Proper. i. e. the measurement of angles by angular units.

Part II. Goniometry by Ratios. i. e. the indication, (not measurement,) of angles by what are called “goniometrical ratios.”

Part III. Trigonometry. i. e. the properties of rectilinear figures.

Before proceeding to enumerate the Formulæ of the subject, I shall make a few remarks on each of these three Parts.

Part I. Goniometry Proper.

Two different units are in common use:

(1). a right angle.

(2). the angle which is subtended, (in any circle), by an arc equal to the radius; this can easily be proved to be the same, whatever be the size of the circle, and to be about $\frac{2}{3}$ of a right angle.

This angle I propose to call “a radial angle.”

When (1) is employed, it is subdivided, either into 90 degrees (English measure), or 100 grades (French measure).

When (2) is employed, the following proportion holds good; $No. of radial angles contained : 1 :: arc : radius,$ which may be briefly expressed thus; $angle = \frac{arc}{radius}$ .

The quantity “2 right angles” would be algebraically represented—in English measure by “180°”—in French measure by 200^g—and in “radial” measure by “3.14159… radial angles.”

Part II. Goniometry by Ratios.

The student will do well to observe that the statement, to be found in most treatises on Trigonometry, that “if any of the quantities, $sin A$ , $cos A$ , &c., be given, the angle A may be determined,” is untrue. The word “angle” is used in various senses; in one of these senses, it could not be determined, even if all the “goniometrical ratios” were given, and even in its simplest sense “ $sin A$ ” is never sufficient to determine it. I believe that the chief difficulty of Trigonometry is surmounted, when we have once arrived at a clear notion of the different meanings of this word “angle.” It may be thought that the following explanation is unnecessarily complicated, but if such complication really exists in the science itself, it surely deserves investigation; we can never get rid of the difficulties of a subject by ignoring them.

First, then, there is the “geometrical angle,” or the angle as treated of by Euclid: of this it may be remarked, that it is considered as an absolute magnitude, without regard to direction, and that it is always measured the shortest way round, so that it can never exceed two right angles. Thus, in the annexed figure, the “geometrical angle” contained between $O X$ and $O A$ is measured (whether from X to A, or from A to X, does not matter) along the arrow marked S, and not along that marked L. If $O X$ and $O A$ were in one straight line, the angle might still be called geometrical, though not treated of in Euclid; it would then be equal to two right angles, and might of course be measured either way round at pleasure.

An angle AOX. Both the shorter angle S and the longer angle L are marked with round arrows.

Secondly, there is the “angle of position”: in measuring this, one of the lines is supposed to be fixed in position, and the angle to be measured, from it, to the other, the shortest way round: and as this angle may lie in either of two opposite directions, these are distinguished as “positive“ and “negative.” Thus, in the annexed figure, the angles $X O A$ and $X O A^{'}$ are identical when viewed as “geometrical angles” only, since they have the same magnitude: nevertheless $O A$ and $O A^{'}$ have different positions with regard to $O X$ , and these may be distinguished by calling the angle $X O A$ “positive,” and $X O A^{'}$ “negative.” Observe also, that every possible position of $O A$ , if above the line $X^{'} O X$ , may be represented by a positive “geometrical angle,” if below it, by a negative one, and if coinciding with $O X^{'}$ , by a positive or negative one, equal to two right angles; but that in no case need the angle exceed two right angles.

$An angle XOA, marked with a round arrow S from X to A. The mirror image XOA’ is marked S’, X’ is the mirror image of X.$

Thirdly, there is the “angle of revolution”: in measuring this, one of the lines is (as before) supposed to be fixed, and the angle to be measured from it to the other; but no longer are we obliged to do this the shortest way round, nor even to stop measuring it the first time we come upon the other line: we may go round and round the circle any number of times, the only rule being that we must at last stop on the other line. Thus, in the annexed figure, the arrows, $S_{1}$ , $S_{2}$ , and $S_{3}$ , are specimens of the various ways in which the “angle of revolution” may be measured from $O X$ to $O A$ ; $S_{1}$ and $S_{3}$ being “positive,” and $S_{2}$ “negative.”

An angle XOA, marked with three round arrows. S1 goes the short way from X to A, S2’ the long way. S3 goes round O two times before then going to A.

Let us now consider how far the “goniometrical ratios” enable us to determine an angle under each of these senses.

It is evident that these ratios depend only on the position of $O A$ with regard to $O X$ , and are unaffected by the method in which we may choose to measure the angle; so that they are of no use in determining the “angle of revolution”; this can only be expressed by referring it to some angular unit.

How far, then, will they help us in determining the “angle of position”? We may see by the annexed figure that if the sine only of $X O A_{1}$ were given, we could not distinguish it from $X O A_{2}$ ; nor, by its cosine only, could we distinguish it from $X O A_{3}$ ; nor, by its tangent only, from $X O A_{4}$ ; and the same may be said of its cosecant, its secant, and its cotangent, which are the reciprocals of the former three. Hence no one of the goniometrical ratios is sufficient by itself to determine an “angle of position”: and since from any one we may determine the magnitude of all the rest, and the magnitude and sign of the reciprocal ratio, we may lay it down as a rule, that in order to determine an “angle of position” we require to know the magnitude of some one ratio, and the signs of two, and that these two must not be reciprocals of each other, (to which we may add, that they must not be the cosine and versed-sine).

A line X’OX with four points around it: A1 top right, A2 top left, A3 bottom right, A4 bottom left.

But in the case of the “geometrical angle,” (as, for instance, when we are treating of one of the angles of a triangle, which is always positive, and can never exceed two right angles,) can this never be determined unless two “goniometrical ratios” are given? In some cases it can: but not in all. If in the last figure we confine our attention to the positive angles only, we see that if the sine only of $X O A_{1}$ were given, we could not distinguish it from $X O A_{2}$ , (and the same may be said of its reciprocal, the cosecant); but that from any one of the remaining ratios the angle might be absolutely determined.

Hence the statement, which I quoted at the beginning of this dissertation, that “if any of the quantities, $sin A$ , $cos A$ , &c., be given the angle A may be determined,” ought to be read thus: “to determine the angle A as a geometrical angle, one of the 4 quantities, $cos A$ , $tan A$ , $sec A$ , $cot A$ , must be given ( $sin A$ and $cosec A$ not being sufficient for this purpose): to determine it as an angle of position, the magnitude of one of these 6 quantities must be given, and the signs of two (which two must not be reciprocals): to determine it as angle of revolution, it must be referred to some angular unit.”

N.B. The phrase “goniometrical ratio,” besides being too long for constant repetition, conveys the false notion of its measuring the angle, (whereas the measure always varies as the thing measured,) instead of merely indicating its value: and does not convey the notion that two of these ratios are generally necessary to determine the angle; for this latter purpose some term analogous to “co-ordinate,” (which, by the way should rather be “co-ordinant,”) is needed. I propose, then, to call them the “co-indicants” of the angle.

In the following Formulæ, the “data” are separated from the “quæsita,” so that, by covering half of the page, the student may test for himself the accuracy of his recollection of them.

The following are the new symbols introduced:

Symbol.	Name.	Meaning.
.	sin.	sine.
.	cos.	cosine.
.	sec.	secant.
.	cosec.	cosecant.
.	tan.	tangent.
.	cot.	cotangent.
.	versin.	versed-sine.

Part III. Trigonometry.

It should be observed that the angles of a rectilinear Figure are considered as angles of absolute magnitude only, i. e. as “geometrical angles.” Hence the angle A can be determined from $cos A$ , or $sec A$ , or $tan A$ , or $cot A$ ; but it can not be determined from $sin A$ , or $cosec A$ : this gives rise to what is called the “ambiguous case” in the solution of Triangles.

Formulæ of Part I.

Given an angle expressed in one of the 3 measures, English, French, and radial; to express it in another
N.B. The number of English degrees in an angle is represented by “`E`”; the number of French grades by “`F`”; the number of radial angles by “ $Θ$ ”; and the number of radial angles contained in two right angles, (i. e. 3.14159&c,) by “`π`.”
(1) Formula connecting `E` and `F`,	$E : F :: 9 : 10$ .
(1) Formula connecting `E` and $Θ$ ,	$E : Θ :: 180 : π$ .

Formulæ of Part II.

I. Formulæ concerning one angle only.
(α) Given magnitude and sign of a co-indicant: to find magnitude and sign of one other.
(1) The pairs of reciprocals are,	and , and , and .
(β) Given magnitude of a co-indicant: to find magnitude of the rest.
(2) Formula connecting and ,	$^{2} +^{2} = 1$ .
(3) Formula connecting , , and ,	$= \frac{}{}$ .
(4) Formula connecting and ,	$^{2} =^{2} + 1$ .
(5) Formula connecting and ,	$= 1 -$ .
(γ) Given certain angles: to find their co-indicants.
(6) , , and of 270° =	0, 1, 0.
90° =	1, 0, $\frac{1}{0}$ .
180° =	0, −1, 0.
270° =	−1, 0, $\frac{1}{0}$ .
45° =	$\frac{1}{\sqrt{2}}$ , $\frac{1}{\sqrt{2}}$ , 1.
60° =	$\frac{\sqrt{3}}{2}$ , $\frac{1}{2}$ , $\sqrt{3}$ .
30° =	$\frac{1}{2}$ , $\frac{\sqrt{3}}{2}$ , $\frac{1}{\sqrt{3}}$ .
II. Formulæ concerning 2 or more angles.
(α) Given the co-indicants of 2 angles: to find the co-indicants of their sum and difference.
(7) $\bar{A + B} =$	$A . B + A . B$ .
(7) $\bar{A - B} =$	$A . B - A . B$ .
(7) $\bar{A + B} =$	$A . B - A . B$ .
(7) $\bar{A - B} =$	$A . B + A . B$ .
Hence $2 A =$	$2 A . A$ .
Hence $2 A$ , in terms of $A$ and $A$ , =	$^{2} A -^{2} A$ .
Hence $2 A$ , in terms of $A$ only, =	$1 - 2^{2} A$ .
Hence $2 A$ , in terms of $A$ only, =	$2^{2} A - 1$ .
(8) $\bar{A + B} =$	$\frac{A + B}{1 - A . B}$ .
(8) $\bar{A - B} =$	$\frac{A - B}{1 + A . B}$ .
Hence $2 A =$	$\frac{2 A}{1 -^{2} A}$ .
(9) The formulæ of (8) may also be written thus, by taking $^{- 1} t$ to mean “the angle whose tangent is `t`.”
$^{- 1} t_{1} +^{- 1} t_{2} =$	$^{- 1} \frac{t_{1} + t_{2}}{1 - t_{1} t_{2}}$ .
$^{- 1} t_{1} -^{- 1} t_{2} =$	$^{- 1} \frac{t_{1} - t_{2}}{1 + t_{1} t_{2}}$ .
$2^{- 1} t =$	$^{- 1} \frac{2 t}{1 - t^{2}}$ .
(β) Given the co-indicants of 3 angles: to find the co-indicants of their sum.
(10) $\bar{A + B + C} =$	$A . B . C . (A + B + C - A . B . C)$ .
(10) $\bar{A + B + C} =$	$A . B . C . (1 - B . C - C . A - A . B)$ .
Hence $3 A =$	$A . (3 - 4^{2} A)$ .
Hence $3 A =$	$A . (4^{2} A - 3)$ .
(11) $\bar{A + B + C} =$	$\frac{A + B + C - A . B . C}{1 - (B . C + C . A + A . B)}$ .
Hence $3 A =$	$\frac{3 A -^{3} A}{1 - 3^{2} A}$ .
(12) $^{- 1} t_{1} +^{- 1} t_{2} +^{- 1} t_{3} =$	$^{- 1} \frac{t_{1} + t_{2} + t_{3} - t_{1} . t_{2} . t_{3}}{1 - (t_{2} t_{3} + t_{3} t_{1} + t_{1} t_{3})}$ .
Hence $3^{- 1} t =$	$^{- 1} \frac{3 t - t^{3}}{1 - 3 t^{2}}$ .
(γ) Given 2 angles: to reduce the sum or difference of their corresponding co-indicants to one term (for logarithmic computation).
(13) $A + B =$	$2 \frac{A + B}{2} . \frac{A - B}{2}$ .
(13) $A - B =$	$2 \frac{A + B}{2} . \frac{A - B}{2}$ .
(13) $A + B =$	$2 \frac{A + B}{2} . \frac{A - B}{2}$ .
(13) $A - B =$	$- 2 \frac{A + B}{2} . \frac{A - B}{2}$ .
(13) $A + B =$	$\frac{(A + B)}{A . B}$ .
(13) $A - B =$	$\frac{(A - B)}{A . B}$ .
III. Formulæ concerning the powers of co-indicants.
(15) Demoivre’s theorem, viz.: $(θ \pm \sqrt{- 1} . θ)^{n} =$	$n θ \pm \sqrt{- 1} . n θ$ .
Hence, if $2 θ = v + \frac{1}{v}$ ,
then $2 \sqrt{- 1} . θ =$	$v - \frac{1}{v}$ .
$2 n θ =$	$v^{n} + \frac{1}{v^{n}}$ .
$2 \sqrt{- 1} . n θ =$	$v^{n} - \frac{1}{v^{n}}$ .
(16) Formula expressing $^{n} θ$ in terms of $θ$ , $2 θ$ , $3 θ$ , &c. Rule is	let $(2 θ)^{n} = {(v + \frac{1}{v})}^{n}$ .
(17) Formula expressing $^{n} θ$ in terms of $θ$ , $θ$ , $2 θ$ , $2 θ$ , &c. Rule is	let $(2 \sqrt{- 1} . θ)^{n} = {(v - \frac{1}{v})}^{n}$ .
(18) Formula expressing $^{n} θ$ in terms of $θ$ , $2 θ$ , $3 θ$ , &c. Rule is	in Demoivre’s theorem, equate the possible and impossible parts on both sides, and divide one by the other.
IV. Summation of series of co-indicants.
(19) $A + (A + B) + \dots + (A + \bar{n + 1} . B) =$	$\frac{\frac{n B}{2}}{\frac{B}{2}} . (A + \frac{(n - 1) . B}{2})$ .
$A + (A + B) + \dots + (A + \bar{n + 1} . B) =$	$\frac{\frac{n B}{2}}{\frac{B}{2}} . (A + \frac{(n - 1) . B}{2})$ .
Hence $A + 2 A + \dots + n A =$	$\frac{\frac{n A}{2}}{\frac{A}{2}} . \frac{(n + 1) . A}{2}$ .
$A + 2 A + \dots + n A =$	$\frac{\frac{n A}{2}}{\frac{A}{2}} . \frac{(n + 1) . A}{2}$ .
V. Formulæ connecting the co-indicants of an angle with its radial measure.
N.B. The number $(1 + \frac{1}{1} + \frac{1}{2} + \dots ad inf.)$ , i. e. the number 2.71828&c., is represented by “`ϵ`.” Also, when the symbol `θ` is used without any co-indicant symbol, it represents the angle in radial measure.
(20) Formulæ expressing $θ$ and $θ$ in terms of `θ`, $θ^{2}$ , $θ^{3}$ , &c.	$θ = 1 - \frac{θ^{2}}{2} + \frac{θ^{4}}{4} - &c.$
	$θ = θ - \frac{θ^{3}}{3} + \frac{θ^{5}}{5} - &c.$
(21) Gregorie’s series, expressing `θ` in terms of $θ$ , $^{3} θ$ , &c. (i. e. expressing $^{- 1} t$ in terms of `t`, $t^{3}$ , &c.)	$θ = \frac{θ}{1} - \frac{^{3} θ}{3} + \frac{^{5} θ}{5} - &c.$
	i. e. $^{- 1} t = \frac{t}{1} - \frac{t^{3}}{3} + \frac{t^{5}}{5} - &c.$
Hence to find `π`. Rule is	let $θ = \frac{1}{2}$ a right angle.
(22) Euler’s series to find `π`. Rule is	in the formula $^{- 1} t_{1} -^{- 1} t_{2} =^{- 1} \frac{t_{1} - t_{2}}{1 + t_{1} t_{2}}$ , let $t_{1} = 1$ , and $t_{2} = \frac{1}{2}$ , and use Gregorie’s series.
(23) Machin’s series to find `π`. Rule is	in the formula $2^{- 1} t =^{- 1} \frac{2 t}{1 - t^{2}}$ , let $t = \frac{1}{5}$ , multiply both sides by 2, and reduce the right-hand side by the same formula; subtract the equation $(\frac{π}{4} =^{- 1} 1)$ from the result, and use Gregorie’s series.
(24) Formulæ expressing $θ$ and $θ$ in terms of `θ` and `ϵ`.
$θ =$	$\frac{1}{2} . (ϵ^{θ \sqrt{- 1}} + ϵ^{- θ \sqrt{- 1}})$ .
$θ =$	$\frac{1}{2 \sqrt{- 1}} . (ϵ^{θ \sqrt{- 1}} - ϵ^{- θ \sqrt{- 1}})$ .

Formulæ of Part III.

(α) Given certain magnitudes concerning a triangle: to find certain other magnitudes.
N.B. The 3 sides are represented by “`a`, `b`, `c`”; the opposite angles by “`A`, `B`, `C`”; and the quantity $\frac{a + b + c}{2}$ by “`S`.”
(1) “Formula of sines.”	$\frac{A}{a} = \frac{B}{b} = \frac{C}{c}$ .
(2) “Formula of sides.”	$A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$ .
(3) “Formula of tangents.”	$\frac{a - b}{a + b} = \frac{A - B}{2} . \frac{C}{2}$ .
(4) $\frac{A}{2}$ , in terms of the sides, =	$\sqrt{\frac{S . (S - a)}{b c}}$ .
(4) $\frac{A}{2}$ , in terms of the sides, =	$\sqrt{\frac{(S - b) . (S - c)}{b c}}$ .
(5) $A$ , in terms of the sides, =	$\frac{2}{b c} \sqrt{S . (S - a) . (S - b) . (S - c)}$ .
(6) Area of triangle, in terms of the sides, =	$\sqrt{S . (S - a) . (S - b) . (S - c)}$ .
(7) Area of triangle, in terms of 2 sides and the included angle, =	$\frac{b c}{2} . A$ .
N.B. The radius of the circle inscribed in a triangle is represented by “`r`”; the radius of the circumscribed circle by “`R`”; and the radii of the 3 escribed circles, respectively touching the sides “`a`, `b`, `c`,” by “ $R_{a}$ , $R_{b}$ , $R_{c}$ .” Also the quantity $\sqrt{S . (S - a) . (S - b) . (S - c)}$ is represented by `M`.
(8) `r`, in terms of the sides, =	$\frac{M}{S}$ .
(9) `R`, in terms of the sides, =	$\frac{a b c}{4 M}$ .
(10) $R_{a}$ , $R_{b}$ , and $R_{c}$ , in terms of the sides, =	$\frac{M}{S - a}$ , $\frac{M}{S - b}$ , and $\frac{M}{S - c}$ .
(β) Given certain magnitudes connected with a quadrilateral figure, whose opposite angles are supplementary: to find certain other magnitudes.
N.B. The 4 sides are represented by “`a`, `b`, `c`, `d`”; the quantity $\frac{a + b + c + d}{2}$ by “`S`”; and the quantity $\sqrt{(a b + c d) . (a c + b d) . (a d + b c)}$ by “`D`”.
(11) Area of figure, in terms of the sides, =	$\sqrt{(S - a) . (S - b) . (S - c) . (S - d)}$ .
(12) Length of diagonal lying between the sides `a`, `b`, and the sides `c`, `d`, =	$\frac{D}{a b + c d}$ .
(13) Length of diagonal lying between the sides `a`, `d`, and the sides `b`, `c`, =	$\frac{D}{a d + b c}$ .
N.B. The radius of the circumscribed circle is represented by “`R`,” and the quantity $\sqrt{(S - a) . (S - b) . (S - c) . (S - d)}$ by `M`.
(14) `R`, in terms of sides, =	$\frac{D}{4 M}$ .
(γ) Given certain magnitudes connected with a regular polygon: to find certain other magnitudes.
N.B. The length of each side is represented by “`a`,” and the number of the sides by “`n`.”
(15) Area, in terms of `a`, =	$\frac{n a^{2}}{4} . \frac{180°}{n}$ .
N.B. The radius of the inscribed circle is represented by “`r`,” and of the circumscribed by “`R`.”
(16) `r`, in terms of `a`, =	$\frac{a}{2} . \frac{180°}{n}$ .
(17) `R`, in terms of `a`, =	$\frac{a}{2} . \frac{180°}{n}$ .
(18) Area, in terms of `r`, =	$n r^{2} . \frac{180°}{n}$ .
(19) Area, in terms of `R`, =	$\frac{n R^{2}}{2} . \frac{360°}{n}$ .