medicine, an extraordinary illuse or evacuation of some humour. Fluxes are various and variously denominated according to their seats or the humours thus voided; as a flux of the belly, uterine flux, hepatic flux, salival flux, &c. The flux of the belly is of four kinds, which have each their respective denominations, viz. the lientery, or fluxus lientericus; the cæliac, or fluxus chylous; the diarrhoea; and the dysentery, or bloody flux;—all which are properly treated of in Medicine.

hydrography, a regular periodical motion of the sea, happening twice in 24 hours; wherein the water is raised and driven violently against the shores. The flux or flow is one of the motions of the tide; the other, whereby the water sinks and retires, is called the reflux or ebb. There is also a kind of rest or cessation of about half an hour between the flux and reflux; during which time the water is at its greatest height, called high-water. The flux is made by the motion of the water of the sea from the equator towards the poles; which, in its progress, striking against the coasts in its way, and meeting with opposition from them, swells, and where it can find passage, as in flats, rivers, &c. rises up and runs into the land. This motion follows, in some measure, the course of the moon; Flux. As it loses or comes later every day by about three quarters of an hour, or more precisely, by 48 minutes; and by so much is the motion of the moon slower than that of the sun. It is always highest and greatest in full moons, particularly those of the equinoxes. In some parts, as at Mount St Michael, it rises 80 or 90 feet, though in the open sea it never rises above a foot or two; and in some places, as about the Morca, there is no flux at all. It runs up some rivers above 120 miles. Up the river Thames it only goes 80, viz. near to Kingston in Surrey. Above London bridge the water flows four hours and ebbs eight; and below the bridge, flows five hours and ebbs seven.

metallurgy, is sometimes used synonymously with fusion. For instance, an ore, or other matter, is said to be in liquid flux, when it is completely fused.

But the word flux is generally used to signify certain saline matters, which facilitate the fusion of ores, and other matters which are difficultly fusible in effays and reductions of ores. Fixed alkalis, nitre, borax, tartar, and common salt, are the saline matters of which fluxes are generally composed. But the word flux is more particularly applied to mixtures of different proportions of only nitre and tartar; and these fluxes are called by particular names, according to the proportions of these ingredients, as in the following articles.

White Flux, is made with equal parts of nitre and of tartar detonated together, by which they are alkalified. The residuum of this detonation is an alkali composed of the alkalis of the nitre and of the tartar, both which are absolutely of the same nature. As the proportion of nitre in this mixture is more than is sufficient to consume entirely all the inflammable matter of the tartar, the alkali remaining after the detonation is perfectly white, and is therefore called white flux; and as this alkali is made very quickly, it is also called expeditious alkali. When a small quantity only of white flux is made, as a few ounces for instance, some nitre always remains undecomposed, and a little of the inflammable principle of the tartar, which gives a red or even a black colour to some part of the flux: but this does not happen when a large quantity of white flux is made; because then the heat is much greater. This small quantity of undecomposed nitre and tartar which remains in white flux is not hurtful in most of the metallic fusions in which this flux is employed: but if the flux be required perfectly pure, it might easily be disengaged from those extraneous matters by a long and strong calcination, without fusion.

Crude Flux. By crude flux is meant the mixture of nitre and tartar in any proportions, without detonation. Thus the mixture of equal parts of the two salts used in the preparation of the white flux, or the mixture of one part of nitre and two parts of tartar for the preparation of the black flux, are each of them a crude flux before detonation. It has also been called white flux, from its colour; but this might occasion it to be confounded with the white flux above described. The name, therefore, of crude flux is more convenient.

Crude flux is detonated and alkalified during the reductions and fusions in which it is employed; and is then changed into white or black flux, according to the proportions of which it is composed. This detonation produces good effects in these fusions and reductions, if the swelling and extravasation of the detonating matters be guarded against. Accordingly, crude flux may be employed successfully in many operations; as, for instance, in the ordinary operation for procuring the regulus of antimony.

Black Flux. Black flux is produced from the mixture of two parts of tartar and one part of nitre detonated together. As the quantity of nitre which enters into the composition of this flux is not sufficient to consume all the inflammable matter of the tartar, the alkali which remains after the detonation contains much black matter, of the nature of coal, and is therefore called black flux.

This flux is designedly so prepared, that it shall contain a certain quantity of inflammable matter; for it is thereby capable, not only of facilitating the fusion of metallic earths like the white flux, but also of reviving these metals by its phlogiston. From this property it is also called reducing flux; the black flux, therefore, or crude flux made with such proportions of the ingredients as to be convertible into black flux, ought always to be used when metallic matters are at once to be fused and reduced, or even when destructive metals are to be fused, as these require a continual supply of phlogiston to prevent their calcination.

Fluxions;

A Method of calculation which greatly facilitates computations in the higher parts of mathematics. Sir Isaac Newton and Mr Leibnitz contended for the honour of inventing it. It is probable they had both made progress in the same discovery, unknown to each other, before there was any publication on the subject.

In this branch of mathematics, magnitudes of every kind are supposed generated by motion; a line by the motion of a point, a surface by the motion of a line, and a solid by the motion of a surface. And some part of a figure is supposed generated by an uniform motion; in consequence of which, the other parts may increase uniformly or with an accelerated or retarded motion, or may decrease in any of these ways; and the computations are made by tracing the comparative velocities with which the parts flow.

Fig. 1. If the parallelogram ABCD be generated Plate by an uniform motion of the line AB toward CD while it moves from FE towards Fe, while the line BF receives the increment Ff, and the figure will be increased by the parallelogram Fe; the line FE in this case undergoes no variation.

The fluxion of any magnitude at any point is the increment that it would receive in any given time, supposing it to increase uniformly from that point; and as the measures will be the same, whatever the time be, we are at liberty to suppose it less than any assigned time.

The first letters in the alphabet are used to represent... sent invariable quantities; the letters $x$, $y$, $z$, variable quantities; and the same letters with points over them $x'$, $y'$, $z'$ represent their fluxions.

Therefore if $AB = a$, and $BF = x$; $EF$, the fluxion of $BF$, will be $\frac{dx}{dt}$, and $Fe$, the fluxion of $AF$, $\frac{da}{dt}$.

If the rectangle be supposed generated by the uniform motion of $FG$ towards $CD$, at the same time that $HG$ moves uniformly towards $AD$, the point $G$ keeping always on the diagonal, the lines $FG$, $HG$ will flow uniformly; for while $BF$ receives the increment $Ef$, and $HB$ the increment $HK$, $FG$ will receive the increment $bg$, and $HG$ the increment $bg$, and they will receive equal increments in equal successive times. But the parallelogram will flow with an accelerated motion; for while $F$ flows to $f$, and $H$ to $K$, it is increased by the gnomon $KGf$; but while $F$ and $H$ flow through the equal spaces $fm$ $KL$, it is increased by the gnomon $Lgm$ greater than $KGf$; consequently when fluxions of the sides of a parallelogram are uniform, the fluxion of the parallelogram increases continually.

The fluxion of the parallelogram $BHGF$ is the two parallelograms $KG$ and $Gf$; for though the parameter receives an increment of the gnomon $KGf$, while its sides flow to $f$ and $K$, the part $gG$ is owing to the additional velocity wherewith the parallelogram flows during that time; and therefore is not part of the measure of the fluxion, which must be computed by supposing the parameter to flow uniformly as it did at the beginning, without any acceleration.

Therefore if the sides of a parallelogram be $x$ and $y$, their fluxions will be $x' y'$; and the fluxion of the parallelogram $xy + yx$; and if $x = y$, that is, if the figure be a square, the fluxion of $x^2$ will be $2xx$.

Fig. 2. Let the triangle $ABC$ be described by the uniform motion of $DE$ from $A$ towards $B$, the point $E$ moving in the line $DF$, so as always to touch the lines $AC$, $CB$; while $D$ moves from $A$ to $F$, $DE$ is uniformly increased, and the increase of the triangle is uniformly accelerated. When $DE$ is in the position $FC$, it is a maximum. As $D$ moves from $F$ to $B$, the line $FC$ decreases, and the triangle increases, but with a motion uniformly retarded.

Fig. 3. If the semicircle $AFB$ be generated by the uniform motion of $CD$ from $A$ towards $B$, while $C$ moves from $A$ to $G$, the line $CD$ will increase, but with a retarded motion; the circumference also increases with a retarded motion, and the circular space increases with an accelerated motion, but not uniformly, the degrees of acceleration growing less as $CD$ approaches to the position $GF$. When $C$ moves from $G$ to $B$, it decreases with a motion continually accelerated, the circumference increases with a motion continually accelerated, and the area increases with a motion continually retarded, and more quickly retarded as $CD$ approaches to $B$.

The fluxion of a quantity which decreases is to be considered as negative.

When a quantity does not flow uniformly, its fluxion may be represented by a variable quantity, or a line of a variable length; the fluxion of such a line is called the second fluxion of the quantity whose fluxion that line is; and if it be variable, a third fluxion may be deduced from it, and higher orders from these in the same manner: the second fluxion is represented by two points, as $x'$.

The increment a quantity receives by flowing for any given time, contains measures of all the different orders of fluxions; for if it increases uniformly, the whole increment is the first fluxion; and it has no second fluxion. If it increases with a motion uniformly accelerated, the part of the increment occasioned by the first motion measures the first fluxion, and the part occasioned by the acceleration measures the second fluxion. If the motion be not only accelerated, but the degree of acceleration continually increased, the two first fluxions are measured as before; and the part of the increment occasioned by the additional degree of acceleration measures the third; and so on. These measures require to be corrected, and are only mentioned here to illustrate the subject.

**DIRECT METHOD.**

Any flowing quantity being given, to find its fluxion.

**Rule I.** To find the fluxion of any power of a quantity, multiply the fluxion of the root by the exponent of the power, and the product by a power of the same root less by unity than the given exponent.

The fluxion of $x^n$ is $nx^{n-1}x'$; for the root of $x^n$ is $x$, whose fluxion is $x'$; which multiplied by the exponent $n$, and by a power of $x$ less by unity than $n$, gives the above fluxion.

If $x$ receive the increment $x'$, it becomes $x+x'$; raise both to the power of $n$, and $x^n$ becomes $x^n+nx^{n-1}x'+\frac{n(n-1)}{2}x^{n-2}x'^2+\ldots$; but all the parts of the increment, except the first term, are owing to the accelerated increase of $x^n$, and form measures of the higher fluxions. The first term only measures the first fluxion; the fluxion of $a^2+z^2$ is $\frac{1}{2}\times2zz\times a^2+z^2$; for put $a=a^2+z^2$, we have $x=2zz$; and the fluxion of $x^{\frac{3}{2}}$, which is equal to the proposed fluent, is $\frac{3}{2}x^{\frac{1}{2}}x'$; for which substituting the values of $z$ and $x$, we have the above fluxion.

**Rule II.** To find the fluxion of the product of several variable quantities multiplied together, multiply the fluxion of each by the product of the rest of the quantities, and the sum of the products thus arising will be the fluxion sought.

Thus the fluxion of $xy$, is $xy+yx'$; that of $xyz$, is $xyz+xyz+yxz$; and that of $xyzu$, is $xyzu+yxyz+xxuy+yzyux$.

**Rule III.** To find the fluxion of a fraction—From the fluxion of the numerator multiplied by the denominator, subtract the fluxion of the denominator multiplied by the numerator, and divide the remainder by the square of the denominator.

Thus, the fluxion of $\frac{x}{y}$ is $\frac{yx'-xy'}{y^2}$; that of $\frac{x}{x+y}$, is $\frac{x(x+y)-x+y\times x}{(x+y)^2}=\frac{yx'-xy'}{(x+y)^2}$. Rule IV. In complex cases, let the particulars be collected from the simple rules, and combined together.

The fluxion of \( \frac{x^2 y^2}{z} \) is \( \frac{2x^2 y + 2y^2 x x z - x^2 y^2 z}{z^2} \); for the fluxion of \( x^2 \) is \( 2xx \), and of \( y^2 \) is \( 2yy \), by Rule I., and therefore the fluxion of \( x^2 y^2 \) (by Rule II.) \( 2x^2 y + 2y^2 x x z \); from which multiplied by \( z \), (by Rule III.) and subtracting from it the fluxion of the denominator \( z \), multiplied by the numerator, and dividing the whole by the square of the denominator, gives the above fluxion.

Rule V. The second fluxion is derived from the first, in the same manner as the first from the flowing quantity.

Thus the fluxion of \( x^3 \), \( 3x^2 \); its second, \( 6x^2 + 3x^2 \) (by Rule II.) ; and so on; but if \( x \) be invariable, \( x = 0 \), and the second fluxion of \( x^3 = 6x^2 \).

Prob. I. To determine maxima and minima.

When a quantity increases, its fluxion is positive; when it decreases, it is negative; therefore when it is just between increasing and decreasing, its fluxion is \( = 0 \).

Rule. Find the fluxion, make it \( = 0 \), whence an equation will result that will give an answer to the question.

Examp. To determine the dimensions of a cylindric measure ABCD, (fig. 4.) open at the top, which shall contain a given quantity (of liquor, grain, &c.) under the least internal superficies possible.

Let the diameter \( AB = x \), and the altitude \( AD = y \); moreover, let \( p \) (\( 3,14159 \), &c.) denote the periphery of the circle whose diameter is unity, and let \( c \) be the given content of the cylinder. Then it will be \( 1 : p :: x : (px) \) the circumference of the base; which, multiplied by the altitude \( y \), gives \( px y \) for the concave superficies of the cylinder. In like manner, the area of the base, by multiplying the same expression into \( \frac{1}{4} \) of the diameter \( x \), will be found \( = \frac{px^2}{4} \); which drawn into the altitude \( y \), gives \( \frac{px^2 y}{4} \) for the solid content of the cylinder; which being made \( = c \), the concave surface \( px y \) will be found \( = \frac{4c}{x} \), and consequently the whole surface \( = \frac{4c}{x} + \frac{px^2}{4} \). Whereof the fluxion, which is \( -\frac{4cx}{x^2} + \frac{px}{2} \). being put \( = 0 \), we shall get \( -8c \times px^3 = 0 \); and therefore \( x = 2 \sqrt{\frac{c}{p}} \); further, because \( px^2 = 8c \), and \( px^2 y = 4c \), it follows, that \( x = 2y \); whence \( y \) is also known, and from which it appears that the diameter of the base must be just double of the altitude.

Fig. 7. To find the longest and shortest ordinates of any curve, DEF, whose equation or the relation which the ordinates bear to the abscissas is known.

Make AC the abscissa \( x \), and CE the ordinate \( y \); take a value \( y \) in terms of \( x \), and find its fluxion; which making \( = 0 \), an equation will result whose roots give the value of \( x \) when \( y \) is a maximum or a minimum.

To determine when it is a maximum and when a minimum, take the value of \( y \), when \( x \) is a little more than the root of the equation to found, and it may be perceived whether it increases or decreases.

If the equation has an even number of equal roots, \( y \) will be neither a maximum nor minimum when its fluxion is \( = 0 \).

Prob. II. To draw a tangent to any curve.

Fig. 5. When the abscissa CS of a curve moves uniformly from A to B, the motion of the curve will be retarded if it be concave, and accelerated if convex towards AB; for a straight line TC is described by an uniform motion, and the fluxion of the curve at any point is the same as the fluxion of the tangent, because it would describe the tangent if it continued to move equally from that point. Now if \( Sx \) or \( Cx \) be the fluxion of the base, \( Cd \) will be the fluxion of the tangent, and de of the ordinate. And because the triangles FSC, Ced, are equiangular, \( de : ce :: CS : ST \), therefore,

Rule. Find a fourth proportional to the fluxion of the ordinate valued in terms of the abscissa, the fluxion of the abscissa, and the ordinate, and it determines the line ST, which is called the semi-tangent, and TC joined is a tangent to the curve.

Examp. To draw a right line CT, (fig. 6.) to touch a given circle BCA in a point C.

Let CS be perpendicular to the diameter AB, and put \( AB = a \), \( BS = x \), and \( SC = y \); then, by the property of the circle, \( y^2 (CS^2) = BS \times AS (x \times a - x) = ax - x^2 \); whereof the fluxion being taken, in order to determine the ratio of \( x \) and \( y \), we get \( 2yy = ax - 2xx \); consequently

\[ \frac{x}{y} = \frac{2y}{a - 2x} = \frac{y}{\frac{1}{2}a - x} \]

which multiplied by \( y \), gives \( \frac{y^2}{\frac{1}{2}a - x} = \) the subtangent ST. Whence (O being supposed the centre) we have \( OS (\frac{1}{2}a - x) : CS (y) :: CS (y) : ST \); which we also know from other principles.

Prob. III. To determine points of contrary flexure in curves.

Fig. 7. Supposing C to move uniformly from A to B, the curve DEF will be convex towards AB when the celerity of E increases, and concave when it decreases; therefore at the point where it ceases to be convex and begins to be concave, or the opposite way, the celerity of E will be uniform, that is, CE will have no second fluxion. Therefore,

Rule. Find the second fluxion of the ordinate in terms of the abscissa, and make it \( = 0 \); and from the equation that arises you get a value of the abscissa, which determines the point of contrary flexure.

Ex. Let the nature of the curve ARS be defined by the equation \( ay = \frac{1}{2}x^2 + xx \) (the abscissa AF and the ordinate FG being, as usual, represented by \( x \) and \( y \) respectively). Then \( y \), expressing the celerity of the point \( r \), in the line FH, will be equal to \( \frac{\frac{1}{2}a \times x + 2xx}{a} \):

Whose fluxion, or that of \( \frac{1}{2}a \times x + 2xx \) (because \( a \)) and \( x \) are constant) must be equal to nothing; that is, \(-\frac{3}{4}a^2x - \frac{1}{2}x + 2x = 0\): Whence \( a^2x - \frac{1}{2}x = 8 \), \( a^2 = 8x^2 \).

\( 64x^2 = a^2 \); and \( x = \frac{a}{2}AF \); therefore \( FG = \left( \frac{a^2}{4}x^2 + ax \right) \)

\( = \frac{a^2}{4}a \): From which the position of the point \( G \) is given.

**Prob. IV. To find the radii of curvature.**

The curvature of a circle is uniform in every point, that of every other curve continually varying; and it is measured at any point by that of a circle whose radius is of such a length as to coincide with it in curvature in that point.

All curves that have the same tangent have the same first fluxion, because the fluxion of a curve and its tangent are the same. If it moved uniformly on from the point of contact, it would describe the tangent. And the deflection from the tangent is owing to the acceleration or retardation of its motion, which is measured by its second fluxion: and consequently two curves which have not only the same tangent, but the same curvature at the point of contact, will have both their first and second fluxions equal. It is easily proven from thence, that the radius of curvature is

\[ \frac{x^3}{y} \]

\( x, y, \) and \( z \) represent the abscissa, ordinate,

and curve respectively.

**Examp. I.** Let the given curve be the common parabola, whose equation is \( y = a^2x^2 \): Then will \( y = \frac{1}{2}a^2xx - \frac{1}{2}x^3 \),

\[ = \frac{a^2}{2}x^2 \], and (making \( x \) constant) \( y = \frac{1}{2}a^2x^2x - \frac{1}{2}x^3 = \frac{a^2}{2}x^2 \):

\[ = \frac{a^2}{4}x^2 : \text{ Whence } \left( \sqrt{x^2 + y^2} \right) = \frac{x}{2} \sqrt{\frac{4x^2 + a^2}{x}} \]

and the radius of curvature \( \frac{a^2}{2}x^2 \): Which at the vertex, where \( x = 0 \), will be \( \frac{a^2}{2} \).

**Inverse Method.**

From a given fluxion to find a fluent.

This is done by tracing back the steps of the direct method. The fluxion of \( x \) is \( x \); and therefore the fluent of \( x \) is \( x \); but as there is no direct method of finding fluents, this branch of the art is imperfect. We can assign the fluxion of every fluent; but we cannot assign the fluent of a fluxion, unless it be such a one as may be produced by some rule in the direct method from a known fluent.

**General Rule.** Divide by the fluxion of the root, add unity to the exponent of the power, and divide by the exponent so increased.

For, dividing the fluxion \( nx^{n-1}x \) by \( x \) (the fluxion of the root \( x \)) it becomes \( nx^{n-1} \); and, adding 1 to the exponent \( (n-1) \), we have \( nx^n \); which, divided by \( n \), gives \( x^n \), the true fluent of \( nx^n \).

Hence (by the same rule) the Fluent of \( 3x^2x \) will be \( x^3 \);

That of \( 8x^2x = \frac{8x^3}{3} \);

That of \( 2x^2x = \frac{x^3}{3} \);

That of \( y^2y = \frac{y^3}{3} \).

Sometimes the fluent so found requires to be corrected. The fluxion of \( x \) is \( x \); and the fluxion of \( a+x \) is also \( x \); because \( a \) is invariable, and has therefore no fluxion.

Now when the fluent of \( x \) is required, it must be determined, from the nature of the problem, whether any invariable part, as \( a \), must be added to the variable part \( x \).

When fluents cannot be exactly found, they can be approximated by infinite series.

**Ex.** Let it be required to approximate the fluent of

\[ \frac{a^2 - x^2}{c^4 - x^2} \times \frac{n}{x} \]

in an infinite series.

The value of \( \frac{a^2 - x^2}{c^4 - x^2} \), expressed in a series, is \( \frac{a}{c} + \)

\[ \frac{1}{2c^3 - 2ac} \times x^2 + \frac{3}{8c^5 - 4ac^3} \times x^4 + \frac{5a}{16c^7 - 16ac^5} \times x^6 + &c. \]

Which value being therefore multiplied by \( x^n \), and the fluent taken (by the common method) we get \( \frac{ax^{n+1}}{n+1} + \frac{a}{2c^3 - 2ac} \times \frac{x^{n+3}}{n+3} + \)

\[ \frac{3a}{8c^5 - 4ac^3} \times \frac{x^{n+5}}{n+5} + \frac{5a}{16c^7 - 16ac^5} \times \frac{x^{n+7}}{n+7} + &c. \]

**Prob. I. To find the area of any curve.**

**Rule.** Multiply the ordinate by the fluxion of the abscissa, and the product gives the fluxion of the figure, whose fluent is the area of the figure.

**Examp. I.** Fig. 8. Let the curve ARMH, whose area you will find, be the common parabola. Let \( u \) represent the area, and \( v \) its fluxion.

In which case the relation of \( AB(x) \) and \( BR(y) \) being expressed by \( y = ax \) (where \( a \) is the parameter) we hence get \( y = a^2x^2 \); and therefore \( u = RmHB = \frac{1}{2}yx \)

\[ = \frac{1}{2}a^2x^2 \]: whence \( u = \frac{1}{2}a^2x^2 \times \frac{1}{2}a^2x^2 \times x = \frac{1}{2}yx \) (because \( a^2x^2 = y \)) \( = \frac{1}{2}AB \times BR \): hence a parabola is \( \frac{1}{2} \) of a rectangle of the same base and altitude.

**Examp. 2.** Let the proposed curve CSDR (fig. 9.) be of such a nature, that (supposing \( AB \) unity) the sum of the areas CSTBC and CDGBC answering to any two proposed abscissas AT and AG, shall be equal to the area CRNBC, whose corresponding abscissa AN is equal equal to \( AT \times AG \), the product of the measures of the two former abscissas.

First, in order to determine the equation of the curve (which must be known before the area can be found), let the ordinates \( GD \) and \( NR \) move parallel to themselves towards \( HF \); and then having put \( GD = y \), \( NR = z \), \( AT = a \), \( AG = s \), and \( AN = u \), the fluxion of the area \( CDGB \) will be represented by \( yz \), and that of the area \( CRNB \) by \( zu \); which two expressions must, by the nature of the problem, be equal to each other; because the latter area \( CRNB \) exceeds the former \( CDGB \) by the area \( CSTB \), which is here considered as a constant quantity; and it is evident, that two expressions, that differ only by a constant quantity, must always have equal fluxions.

Since, therefore, \( yz = zu \), and \( u = as \), by hypothesis, it follows, that \( u = ai \), and that the first equation (by substituting for \( u \)) will become \( y = axz \), or \( y = az \), or lastly \( y = za \), that is, \( GD \times AG = NR \times AN \); therefore, \( GD : NR :: AN : AG \); whence it appears, that every ordinate of the curve is reciprocally as its corresponding abscissa.

Now, to find the area of the curve so determined, put \( AB = 1 \), \( BC = b \), and \( BG = x \); then, since \( AG = (1 + x) : AB (1) :: BC (b) : GD (y) \) we have \( y = \frac{b}{1 + x} \),

consequently \( u = \frac{b^2}{1 + x} = b \times x - x^2 + x^3 - x^4 + \ldots \)

Whence, \( BGD \), the area itself will be \( b \times x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} + \ldots \), &c. which was to be found.

Hence it appears, that as these areas have the same properties as logarithms, this series gives an easy method of computing logarithms; and the fluent may be found by means of a table of logarithms, without the trouble of an infinite series; and every fluxion whose fluent agrees with any known logarithmic expression, may be found the same way. Hence the fluents of fluxions of the following forms are deduced.

The fluent of \( \frac{x}{\sqrt{x^2 + a^2}} = \text{hyp.log. of } x + \sqrt{x^2 + a^2} \);

of \( \frac{x}{\sqrt{2ax + x^2}} = \text{hyp.log. } ax + \sqrt{2ax + x^2} \);

of \( \frac{2ax}{a^2 - x^2} = \text{hyp.log. of } \frac{a + x}{a - x} \);

and of \( \frac{2ax}{x\sqrt{a^2 - x^2}} = \text{hyp.log. } a - \sqrt{a^2 - x^2} \).

**Prob. 2. To determine the length of curves.**

Fig. 5. Because \( Cde \) is a right-angle triangle, \( Cd^2 = Ce^2 + de^2 \); wherefore the fluxions of the abscissa and ordinate being taken in the same terms and squared, their sum gives the square of the fluxion of the curve; whose root being extracted, and the fluent taken, gives the length of the curve.

**Examp.** To find the length of a circle from its tangent. Make the radius \( AO \) (fig. 5.) \( = a \), the tangent of \( AC = t \), and its secant \( = s \), the curve \( = z \), and its

fluxion \( = \dot{z} \); because the triangles \( OTC, OCS \) are similar, \( OT : OC :: OC : OS \); whence \( OS = \frac{a^2}{s} \), and \( SA = a - \frac{a^2}{s} = a - \frac{a^2}{\sqrt{a^2 + t^2}} \); whose fluxion is \( \frac{a^2t}{a^2 + t^2} \); and because the triangles \( OTC, dCe \) are similar, \( TC (= t) : TO (= \sqrt{a^2 + t^2}) :: Ce = \frac{a^2t}{a^2 + t^2} \); \( Cd = \frac{a^2t}{a^2 + t^2} \) fluxion of the curve.

Now by converting this into an infinite series we have the fluxion of the curve \( = \frac{t^2}{a^2} + \frac{t^4}{a^4} - \frac{t^6}{a^6} + \ldots \), &c. and consequently \( z = \frac{t^3}{3a^3} + \frac{t^5}{5a^5} - \frac{t^7}{7a^7} + \frac{t^9}{9a^9} + \ldots \), &c. \( = AR \).

Where, if (for example's sake) \( AR \) be supposed an arch of 30 degrees, and \( AO \) (to render the operation more easy) be put unity, we shall have \( t = \sqrt{\frac{1}{3}} = .5773502 \) (because \( Ob \times \frac{1}{4} : LR (\frac{1}{4}) :: OA (1) : AT (\frac{1}{4}) = \sqrt{\frac{1}{3}} \))

Whence,

\[ \begin{align*} t^3 &= \left(\frac{1}{3}\right)^3 = .0124500 \\ t^5 &= \left(\frac{1}{3}\right)^5 = .0041500 \\ t^7 &= \left(\frac{1}{3}\right)^7 = .0013833 \\ t^9 &= \left(\frac{1}{3}\right)^9 = .00071277 \\ t^{11} &= \left(\frac{1}{3}\right)^{11} = .00023759 \\ t^{13} &= \left(\frac{1}{3}\right)^{13} = .00007919 \\ t^{15} &= \left(\frac{1}{3}\right)^{15} = .00002639 \\ &\vdots \end{align*} \]

And therefore \( AR = .5773502 - .01934500 + .0041500 - .0013833 + .00071277 - .00023759 + .00007919 - .00002639 + \ldots \)

\( = .5735987 \): for the length of an arch of 30 degrees, which multiplied by 6 gives \( 3.41592 \) for the length of the semi-periphery of the circle whose radius is unity.

Other series may be deduced from the versed sine and secant; and these are of use for finding fluents which cannot be expressed in finite terms.

\[ \begin{align*} \text{Versed sine} &= \frac{w}{\sqrt{2aw - aw^2}} \\ \text{Right fine} &= \frac{w}{\sqrt{a^2 - aw^2}} \\ \text{Tangent} &= \frac{aw}{a^2 + aw^2} \\ \text{Secant} &= \frac{aw}{w\sqrt{aw^2 - a^2}} \end{align*} \]

R r 2 Prog.