AEROSTATION

, pa. 45, col. 2, l. 40, for 800, read 680.—l. 46, for 28 1/3 read 26.—l. 48, for balloon read parachute.—l. 51 and 52, for 28 1/3 read 26, and for 13 read 12.—l. 55, read 2 feet 3 inches. |

Pa. 46, col. 2, at the end of the article on Aerostation, add, See an ingenious and learned treatise on the mathematical and physical principles of Airballoons, by the late Dr. Damen, professor of philosophy and mathematics in the University of Leyden, entitled, Physical and Mathematical Contemplations on Aerostatic Balloons, &c; in 8vo, at Utrecht, 1784.

Pa. 70, col. 1, l. 4, dele -√(3 - 1 = 2).-l. 5, at the end add -√(3 - 1) = 2.

Pa. 71, col. 1, l. 9, for y² + 2y - 7 read ―(y² + 2y-7).

AFFECTED Equations, add (from Francis Maseres, Esq.)—“This expression of Affected Equations seems to require some further explanation. It was introduced by the celebrated Vieta, the great father and restorer of Algebra. He has many expressions peculiar to himself, and which have not been adopted by subsequent Algebraists. Amongst these are the following ones. He calls a set of quantities in continual geometrical proportion, (such as the quantities 1, x, x², x³, x⁴, x⁵, x⁶ x⁷, &c,) a set of scalar quantities, or magnitudines scalares; and, when there are several of these scalar quantities mentioned together, (as in the compound quantity x⁵ + ax⁴-b²x³,) he calls the highest quantity, or that which is farthest in the scale of quantities 1, x, x², x³, x⁴, x⁵, x⁶, x⁷, &c. (to wit, the quantity x⁵ in the said compound quantity x⁵ + ax⁴-b²x³,) the power of the fundamental quantity x, or of the second term in the said scale; and he calls the lower scalar quantities which are involved in the second and third terms of the said compound quantity x⁵+ax⁴ -b²x³, to wit, the quantities x⁴ and x³, (or, in our present language, the inferior powers of x,) scalar quantities of a parodic degree to x⁵, or the power of the fundamental quantity x. This word parodic I take to be derived (though Vieta does not tell us so) from the Greek words pxrx\ and o(do\s, which signify near and a way or road, because these inferior scalar quantities x³ and x⁴ lie in the way as you pass along in the scale of the aforesaid quantities 1, x, x², x³, x⁴, x⁵, x⁶ x⁷, &c, from 1 to x⁵, which he calls the power of x in the said compound quantity x⁵ + ax⁴-b²x³. These inferiour scalar quantities x³ and x⁴ are therefore parodic, or situated in the way to, or are leading to, the higher scalar quantity x⁵. He then proceeds to define a pure power and an affected power, and tells us that a pure power is a scalar quantity that is not affected with any parodic, or inferiour scalar quantity, and that an affected power is a scalar quantity that is connected by addition, or subtraction with one, or more, inferiour, or parodic, scalar quantities, combined with co-efficients that raise them to the same dimension as the power itself, or make them homogeneous to it, and consequently capable of being added to it, or subtracted from it. Thus x⁵ alone is a pure power of x, namely, its fifth power; and x⁵ + ax⁴-b²x³ is an affected power of x, namely, its fifth power affected by, or connected with, the two parodic, or inferiour, scalar quantities x³ and x⁴, which are multiplied into bb and a, in order to make them homogeneous to, or of the same dimension with, x itself, and capable of being added to it or subtracted from it. See Schooten's Edition of Vieta's works, published at Leyden in Holland in the year 1646, pages 3 and 4.

“This, then, being the meaning of the expression, a pure power and an affected power, the meaning of the corresponding expressions of a pure equation and an affected equation follows from it of course: a pure equation signifying an equation in which a pure power of an unknown quantity is declared to be equal to some known quantity; such as the equation ; and an affected equation signifying an equation in which a power of an unknown quantity affected by, or connected, either by addition or subtraction, with, some inferiour powers of the same unknown quantity, (multiplied into proper co-efficients in order to make them homogeneous to the said highest power of the said unknown quantity,) is declared to be equal to some known quantity; such as the equation . This I take to be the original meaning of the expression an affected equation. But, as the language of Vieta has not been adopted by subsequent writers of Algebra, I should think it would be more convenient to call them by some other name. And, perhaps those of binomial, trinomial, quadrinomial, quinquinomial, and, in general, that of multinomial equations, would be as convenient as any. Thus, , might all be called binomial equations, because they would be equations in which a binomial quantity, or quantity consisting of two terms that involved the unknown quantity x, is declared to be equal to a known quantity; and, for a like reason, the equations , might be called trinomial equations. And the like names might be given to equations of a greater number of terms. Dr. Hutton, I observe, in his excellent new Mathematical and Philosophical Dictionary, just now published, (Feb. 2. 1795,) calls them compound equations; which is likewise a very proper name for them, and less obscure than that of affected equations.”

Pa. 76, col. 1, l. 25, for √(3 + 1) - √(3 - 1), read √(3 + 1) - √(3 - 1).

Pa. 94, col. 2, l. 34, for Spaniard, read Portuguese.

Pa. 95, col. 2, after l. 21, or the end of the paragraph relating to Dr. Barrow, add as follows:—Of these lectures, the 13th deserves the most special notice, being entirely employed upon Equations, delivered in a very curious way. He there treats of the nature and number of their roots, and the limits of their magnitudes, from the description of lines accommodated to each, viz, treating the subject as a branch of the doctrine of maxima and minima, which, in the opinion of some persons, is the right way of considering them, and far preferable to the so much boasted invention of the generation of Equations from each other discovered by Harriot and Descartes.

Pa. 97, col. 2, after l. 3, add—Dr. Waring and the Rev. M. Vince, of Cambridge, have both given many | improvements and discoveries in series and in other branches of analysis. Those of Mr. Vince are chiefly contained in the latter volumes of the Philosophical Transactions; where also are several of Dr. Waring's; but the bulk of this gentleman's improvements are contained in his separate publications, particularly the Meditationes Algebraicæ, published in 1770; the Proprietates Algebraicarum Curvarum, 1772; and the Meditationes Analyticæ, 1776; an account of the chief contents of which, a friend has favoured me with, as follows.

The first chapter treats of the transformation of algebraical equations into others, of which the roots have given algebraical relation to the roots of the given equations.

The general resolution of this problem requires the finding the aggregates of each of the values of algebraical functions of the roots of the given equation: for this purpose the author begins with finding the sum of the m^th power of each of the roots of the equation by a series proceeding according to the dimensions of p the sum of the roots: this series (when continued in infinitum and converges) finds also the sum of any root of the above-mentioned quantities. From this series is deduced the law of the reversion of the series , which finds x in terms of y; and also the law of a series, which expresses the greatest or least roots, and their powers or roots of a given algebraical equation, and which may be applied whether that root is possible or impossible, if the root be much greater or less than each of the remaining ones. All the powers and roots of this series, when continued in infinitum, observe the same law.

On this subject are further added some elegant theorems; of which, one finds the sum of all quantities of this kind a^ab^bg^c, &c; where a, b, g, &c, denote the roots of the given equation. This has been since published by the celebrated mathematician Mr. le Grange in the Academy of Sciences at Paris.

There is also added a method of considerable utility in these matters; viz, the assuming equations whose roots are known, and thence deducing the coefficients of the equations sought: and also from the terms of an inferior equation deducing the terms of a superior.

The second chapter principally treats of the limits and number of impossible and affirmative and negative roots of algebraical equations.

Some new properties are added, of the limiting equations resulting from multiplying the successive terms of the given equation into an arithmetical series; and a method of finding limits between each of the roots of a given equation, since published in the Berlin Acts, and also some new methods of finding equations whose roots are limits between the roots of other equations. In theor. 4 and 5 are contained quantities which are always greater than certain others, when they are all possible; from whence may be deduced Newton's and several other rules for finding the number of impossible roots: these rules may be rendered somewhat more general by multiplying the given equations into others, whose roots are all possible, and finding whether im- possible roots may be deduced by the rule in the resulting equation, which cannot from it be discovered in the given one. A rule is given, deduced from each successive four terms of the given equation, and consequently much more general than rules deduced from each successive three terms. The former always discovers the true number of impossible roots contained in quadratic and cubic equations, the latter in quadratic only. There is also a rule given for finding the number of impossible roots from an equation, of which the roots are the squares, &c, of the roots of a given equation; and a second from an equation of which the roots are the squares of the differences of the roots of a given equation; and a third rule for finding an equation, of which the root is ; if be the given equation, &c, these latter resolutions always discover the true number of impossible roots contained in cubic, biquadratic and sursolid equations; and also whether or not any impossible roots are contained in any given equation; and also from the last term whether the number of impossible roots contained be 2, 6, 10, &c, or 0, 4, 8, &c. The principle of a 4th rule is given by finding when two roots once, twice, thrice, &c, or four, &c, roots become equal. From a method given of finding the number of impossible roots contained in an equation involving only one unknown quantity, is deduced a method of discovering limits between which are contained any number of impossible roots in an equation involving two or more unknown quantities. From the number of impossible, affirmative and negative roots contained in a given equation, is delivered a method of finding the number of impossible, &c roots contained in an equation of which the roots have a given algebraical relation to the roots of the given equation.

The principles are subjoined of finding the number of affirmative and negative roots contained in an algebraical equation: but this necessarily supposes a method of finding the number of its impossible roots known. It is demonstrated, that if the equation be multiplied by x - a, then every change of signs in the given, will have one, or three, or five, &c in the resulting equation; and if it be multiplied by x + a, then every continuation from + to + or - <*>o -, will produce one, or three, or five, &c such continuations in the resulting, whence every equation will contain at least so many changes of signs in its successive terms as there are affirmative roots, and so many continued progresses from + to + and - to -, as there are negative. In a biquadratic , of which two roots are impossible, and s an affirmative quantity, then it is demonstrated that the two possible ones will be both negative or both affirmative, according as p³ - 4pq + 8r is an affirmative or negative quantity, if the signs of the coefficients, p, q, r, s are neither all affirmative, nor alternately - and +. The number of impossible and affirmative and negative roots contained in the equation is likewise given, &c. If , then the content of all the values | of the quantity w will be to the content of all the values of the quantity v :: ± lⁿ : h^m, from whence are deduced some properties of parabolic curves. Ex. gr. Let the equation expressing the relation between the absciss x and ordinate y be . Then will the content under the (n - 1) greatest ordinates be to the square of the content of all the distances between any two points in which the absciss cuts the curve :: a^n-1 : nⁿ-2. The quotient of the content of all the sines divided by the content of all the cosines to the points in which the absciss cuts the curve, will be to the content of all the abovementioned greatest ordinates :: nⁿa : 1. Similar propositions are deduced concerning the ordinates to the points of contrary flexure, &c.

The third chapter is versant, concerning, 1st finding the roots of equations or irrational quantities, which have given relations to each other: this is performed by substitution or division and finding the common divisors of the quantities resulting; and 2d concerning more (n) equations containing a less number (m) of supposed unknown quantities, which consequently require n-m equations, since named equations of condition; these are likewise deduced from the method of finding common divisors. 3dly, Concerning the resolution of equations; in this case is given, 1. The reduction or resolution of some recurring equations. 2. Some properties of the roots of the equation . 3. Resolution of a biquadratic , by reducing it to an equation . 4. A resolution of the biquadratic by adding (p² + 2n) x² + 2pnx + n² to both sides of the equation, so as to complete the square; and the deducing that the values of n are (ab + gd)/2, (ag + bd)/2, (ad + bg)/2; the values of √ (q + p² + 2n) are (a + b - g - d)/2, (a + g - b - d)/2, &c, and the values of √ (s + n²) are (ab - gd)/2, (ag - bd)/2, &c; if a, b, g, d, are the roots of the given equation. 5. A resolution of equations as general as any yet discovered, viz, the assuming ; and exterminating the irrational quantities, viz, from assuming are deduced different resolutions of cubic; from different resolutions of biquadric; from the equations , are deduced De Moivre's equation, and several others of new formula not before delivered. 6. The resolution , first given by Euler, shewn to be a very particular; but this is rendered here much more general by assuming a more general resolution. 7. The resolution and reduction of equations from exterminating irrational quantities. 8. Reduction of some equations, when they are deduced from others by reducing them to the original equations. 9. The finding a quantity, which multiplied into a given irrational will produce a rational quantity, and thence deducing from a given equation involving irrational quantities the dimensions to which the equation freed from them will ascend. 10. Let P = a series either ascending or descending according to the dimensions of x, from thence is deduced the sum of a series consisting of its alternate terms, or terms at (n) distance from each other. 11. It is proved, that Cardan's resolution of a cubic, is a resolution of an equation of 9 dimensions or three different cubics: similar principles are applied to some other equations. 12. General principles are given for the deducing the function of the roots of the given, which constitute the coefficients or roots of the transformed equation. E. g. Let a cubic equation , thence is shewn the function of the roots of x, which constitute z, and further the cases of the cubic, which are resolvable by the transformed equation, whose root is z: the same principles are applied to biquadratics. 13. The correspondent impossible roots of a given irrational quantity are deduced; and also the different roots of a given resolution. 14. The biquadratic of the formula is distinguished into two quadratic equations involving only possible quantities, and thence every algebraic equation is proved to consist of simple and quadratic divisors involving only possible quantities. 15. A method is delivered of transforming irrational quantities into others; but it is cautioned, that in reduction and transformation correspondent roots should be used, otherwise it is probable that we shall fall into errors, of which examples are given. 16. The convergency of a root found by the common method of approximations is given; and it is discovered that the convergency principally depends on the quantity assumed for the root being much more near to one root than to any other; and independent of it, not on how near it is to a root.

The fourth chapter is principally conversant concerning more algebraical equations and their reductions to one. 1. It gives the law of the resolution of any number of simple equations; and the reduction of n simple equations to n - 1 by means of others. 2. The method of reducing more (n) equations into one so as to exterminate n - 1 unknown quantities by the method of common divisors, and further delivers the principles of investigating the roots or values of the unknown quantities, which result from this, or, which is much the same, from the common method of Erasmus Bartholinus, and which are not contained in the given equations. 3. If two algebraical equations of n and m dimensions of the unknown quantities x and y are reduced to one so as to exterminate one of the unknown quantities, the principles are given of finding the dimensions to which the other will ascend: if it ascends to n X m dimensions; then the sum of the roots depends on the terms of n and n - 1 dimensions in the one, and m and m - 1 in the other, and similarly of the products of every two; &c. From this principle are deduced several properties of algebraical curves. | The same principles are applied to more equations involving more unknown quantities. 4. Some two equations of given formulæ are reduced to one so as to exterminate one unknown quantity. 5. Two equations are likewise reduced to one so as to exterminate unknown quantities by means of insinite series. 6. A method of finding whether some equations contain the same roots of the unknown quantities as others. 7. From the correspondent roots of the unknown quantities in given equations are found the constitution of their coefficients; and from thence the aggregates of the functions of the roots of two or more equations. 8. Some things are given concerning the transformations of more equations than one, of their impossible roots, of their roots which have a given relation to each other. 9. Some reductions and resolutions of more equations involving more unknown quantities. 10. If two equations similarly involve two unknown quantities x and y; then the equation of which the root is x or y is demonstrated to have twice the dimensions of the equation whose root is any rational function of x + y or x² + y² or any rational recurring function of x and y; and if for y be substituted - y; then in the equation whose root is the resulting quantity the dimensions will be the same as in the equations whose root is x or y, but its formula will be of half the number of dimensions. The same principles are applied to more equations similarly involving more unknown quantities. 11. If there are two equations involving two unknown quantities, one deduced from the other, by some substitutions investigated from equations similarly involving two unknown quantities; then the equation whose root is one of the unknown quantities will be recurring. 12. Let A and B be functions of x and y, a method is given of finding, whether A is a function of B. 13. Methods of approximations to the roots of equations when they are unequal, or two or more nearly equal, possible or impossible; and also some remarks on the increments or decrements of the roots, in passing from one equation to others of the same number of dimensions are given.

The fifth chapter treats of rational and integral values of the unknown quantities of given equations.

1. It finds the rational and integral simple, quadratic, &c divisors (by a method different to Waessaner's) of a given equation, which involves one or more unknown quantities. 2. If two equations involve two unknown quantities x and y; the same irrationality, which is contained in x will likewise be contained in its correspondent value of y, unless two or more values of the quantity (x or y) are equal, &c. 3. A method is given of finding integral correspondent values of the unknown quantities of two or more equations involving as many unknown quantities. 4. A method is also delivered of deducing when a given equation can be resolved by means of square, cube, &c roots; and when by similar methods it can be reduced to equations of 1/2, 1/4, &c, its dimensions. 5. A method is given of finding a quantity or number, in which are contained all the divisors of any given rational or integral quantities. 6. A method different from Schooten's, Newton's, and Euler's, of extracting the root of a binomial surd a + √b is given, and the principle demonstrated on which all the rules are founded given by Schooten, viz, the multiplying the binomial surd so that the n^th root of A² - B can be extracted, where A + √B is the resulting surd; and it is further proved that multiplying the given surd a + √b into 2ⁿ will render Newton's resolution as general as the others; and lastly the extraction of the (m^th) root of the quantity A + B√ⁿp + C√ⁿ(p²) + ... + √ⁿ(p^n-1) is given. 7. The law of Dr. Wallis's approximations in terms of the successive quotients, as also of continual fractions is deduced. 8. A method of deducing the integral values of each of the unknown quantities x, y, z, v, &c, contained in the equation in terms of quantities, for which may be assumed any whole numbers. 9. Two or more equations are reduced to one, so as to exterminate unknown quantities; and if the unknown quantities of the resulting equations be integral or fractional, then the unknown quantities of the given equations will also be integral or fractional. 10. Principles are delivered of deducing equations of which the unknown quantities admit of correspondent and known integral or rational values. 11. Correspondent integral or rational values of the unknown quantities in several equations are given, and from some values of the abovementioned kind given, are deduced others. 12. A method of denoting any numbers either by fours, fives, fixes, &c, and their powers; and similar properties deduced as in decimal arithmetic. 13. It is demonstrated that the sum of the divisors of the number 1. 2. 3 ... x = N has to N a greater ratio than the sum of the divisors of any number L less than N has to L; and some other similar properties. 14. In the Philosophical Transactions are given properties similar to Mr. Euler's of the sum of divisors of the natural numbers, and some others. 15. Let , where a, b, r, p and q are whole numbers, then N2m + 1 and N2m + 2 can be compounded by (m + 1) different ways of the quantities p² + rq²; the different ways were first given in the Medit. 16. Every number consists of 1, 2, 3 or 4 squares, and of 1, 2, 3, 4, .. 9 cubes, and therefore if a number N is equal to 3 squares or 8 cubes, the problem may not be possible. 17. Let x and z be any whole numbers, and a and b numbers prime to each other, then ax + bz can constitute any number, which exceeds a X b - a - b. 18. Let r the greatest common divisor of m and n - 1, where n is a prime number; the number of remainders from the division of the number 1^m, 2^m, 3^m, &c, in infinitum by n will be (n - 1)/r + 1: from which are deduced several propositions. 19. Sir John Wilson's property delivered and demonstrated, viz, 1. 2. 3 ... n - 1 + 1 will be divisible by n, if n be a prime number. 20. The sum of the powers 1^r + 2^r + 3^r + ... x^r are found divisible by x. ―(x + 1), if r be a whole number; from whence is deduced an elegant property of all parabolas correspondent to the property of Archimedes of the inscribed triangles in a conical parabola. 21. Some properties of exponential equations; several other new properties of algebraical quantities and equations are given in these Meditations. They were sent to the Royal Socicty in 1757, and since published in the years 1760, 62, and 69, |

The equation expressing the relation between the absciss and its correspondent ordinates of a curve is transformed into another which expresses the relation between different abscissæ and their ordinates, from which is deduced, that there may be n and not more different diameters in a curve of n - 1 order, which cuts its ordinates in a given angle; and likewise that a diameter can have no more than n - 1 different inclinations of its ordinates, unless the diameter be a general one. 2. The formula of the equations to curves, all whose diameters are parallel, or cut each other in a given point, or which have a general diameter to which the lines any how inclined are ordinates. 3. It is proved that there cannot be more than n/m different inclinations of parallel ordinates, which cut the curve in n - m points only, possible or impossible. 4. Something is added concerning diameters, which cut their ordinates on both sides into equal parts. 5. It is demonstrated that there are curves of any number of odd orders, that cut a right line in 2, 4, 6, &c, points only; and of any number of even orders that cut a right line in 3, 5, 7, &c points; and consequently that the order of the curve cannot be denounced from the number of points, in which it cuts a right line. 6. The principles are delivered of finding the asymptotes, parabolical legs, ovals, points, &c, of a curve, of which the equation marking the relation between the absciss and its ordinates is given; and also given the number of asymptotes, parabolical legs of different kinds, ovals, points of different kinds, the least order of a curve, which receives them, is deduced. 7. An equation expressing the relation between an absciss and its ordinates, is transformed into an equation expressing the relation between the distances from two or more points, the latter may be varied an infinite number of ways; and thence are deduced some properties. Many resolutions of this kind are only resolutions of a particular case contained in it; and consequently can never be deduced from any general reasoning; they are often deduced from some particular cases, which are known to answer several conditions of the problem. Transformations of a given curve into others by substitutions, and properties of the loci of some points are deduced, from which Mr. Cotes's property of algebraical curves, and others of a similar and somewhat different nature are derived. 8. Let a curve of n dimensions have n asymptotes, then the content of the n abscissæ will be to the content of the n ordinates, in the same ratio in the curve and asymptotes, the sum of their (n) subnormals to ordinates perpendicular to their abscissæ will be equal to the curve and the asymptotes; and they will have the same central and diametrial curves. 9. Some propositions are added concerning the construction of equations, and some equations are constructed from the principles of Slusius.—If two curves of n and m dimensions have a common asymptote; or the terms of the equations to the curves of the greatest dimensions have a common divisor, then the curves cannot intersect each other in n X m points, possible or impossible. If the two curves have a common general centre, and intersect each other in n X m points, then the sum of the affirmative abscissæ &c to those points will be equal to the sum of the negative; and the sum of the n subnormals to a curve which has a general centre will be proportional to the distance from that centre. 10. Something is added on the description of curves. 11. No curve which has an hyperbolical leg of the conical kind can in general be squared. 12. It is demonstrated that no oval sigure, which does not intersect itself in a given point, can in general be expressed in finite algebraical terms. 13. Given an algebraical equation, and similarly equations expressing a relation between x and y, &c; and also a fluxional quantity which is an algebraical function (z) of x and y and their fluxions; a method is given of deducing an equation whose root is z; and thence some properties of curves. 14. Properties similar to the subsequent of conic sections, are extended to curves of superior orders, viz, if lines be drawn from given points in them in given angles to four lines inscribed in the conic section, then will the rectangle under two of those lines be to the rectangle under the other two in a given ratio. Several properties are added, which follow from the application of algebraical propositions invented in the Medit. Algebr. to curve lines.

The second chapter treats of curvoids and epicurvoids, or curves generated by the rotation of given curves on right lines or curves, and gives a method of rectifying and squaring them; and from the radii of curvature of the generating curves being given, it deduces the length and radius of curvature of the curve generated at the correspondent point; it also asserts that from them may be deduced the construction of the fluxional equations of the different orders.

The third chapter treats of algebraical solids. 1. It deduces the equation to every section of a solid generated by the rotation of a curve round its axis; and from thence the different sections generated by the rotation of conic sections round their axis. 2. The equation to solids contains the relation between the two abscissæ and their ordinates, and the order of the solid may be distinguished according to the dimensions of the equation; or the solid may be defined by two equations expressing the relation between the three abovementioned quantities, and a fourth which may be the axis of the section: there is further given a method of deducing the equation to any section of these solids, and from it the equation to the curve projected on a plane by a given curve. 3. A method of deducing the projection of a curve or solid on each other. 4. If the equation be x - a = 0, (x being the distance from a given point) then it may denote the periphery of a circle if one plane, or the surface of a globe if it refers to a solid. 5. Let x and y denote the distances from two respective points, then an equation expressing the relation between x and y designs the periphery of a curve, if contained in the same plane, or the surface of a solid generated by the rotation of a curve round its axis, passing through the two given points, if a solid. 6. An equation expressing the relation between lines drawn from three or more points may denote an equation to a solid. 7. If x, z and y denote the two abscisses and correspondent ordinates to a solid, and the terms of x and y, or x and z, or y and z; or x, z and y be similarly involved; then may the solid be divided into two | or six similar and equal parts; and if no unequal power of x or y or z; or x and y, &c; or x, y and z be contained in the equation, then the curve may further be divided in general into twice, four or six times the preceding number of equal parts. 7. Curves of double curvature are designed by two equations expressing the relation between two abscissæ and correspondent ordinates, or between lines drawn from three or more points; similar properties may be deduced from these as from the equations to curves.

Chapter the 4th treats of the maxims and minims of polygons inscribed and circumscribed about curves, and thence deduces certain quantities equal to each other, when maxims and minims are contained at every point of the curve: it further contains several properties of conic sections. 1. If any rectilinear figure circumscribes an ellipse, the content under the alternate segments of the line made by the points in which the line touches the ellipse will be equal. 2. If a right line cuts a conic section, and the parts of the line without the conic section on both sides are equal; and any rectilinear figure, which begins and ends at the bounds of the abovementioned line, be described round the conic section, then the contents under the alternate segments of the circumscribing lines as divided in the points of contact will be equal. 3. If two polygons be circumscribed about an ellipse, and the sides are cut by the points of contacts in the same ratios in the one as in the other; then will the areas of the two polygons be equal. 4. If two lines cut a conic section proportionally, i. e. they are divided by the conic section in the same ratio in the one as in the other, and if polygons be described round the conic section, terminated at the ends of those lines, of which the sides are divided by the points of contact in the same ratio in the one as in the other, then will the area of the two polygons be equal, as likewise the curvilinear area. 5. If all the sides of two polygons inscribed in an ellipse make the two angles at the same point equal, and two polygons of this kind be inscribed in the curve, then will the sum of the sides of the one polygon be equal to the sum of the sides of the other. Several other similar properties are added, as also properties of solids generated by the rotation of a conic section round its axis; to which I shall mention the three or four following. 1. The diagonals of a parallelogram circumscribing an ellipse or hyperbola will be conjugate diameters. 2. The sections of a solid generated by the rotation of a conic section round its axis, which pass through its focus, will have that point for the focus of all the sections. 3. If 4 perpendiculars be drawn from any point in an hyperbola to its periphery; and two lines from the same point to the asymptotes and the ordinates from the 4 points of the curve and the 2 of the asymptotes be drawn to the absciss; then will the sum of the resulting abscissæ to the former be double to the sum of the abscissæ to the latter. 4. If an arc of the periphery of a circle be divided into n equal parts, a, 2a, 3a, &c, and p = chord of the arc 180 - na, and a and b be the roots of the quadratic and radius 1: then will aⁿ+bⁿ=chord of the arc 180 - na, from whence may be deduced the divisors of the quantity x²ⁿ - Axⁿ + 1; and also the equation whose roots are the distances of a point in the circle from those points of equal division, and further may be deduced the sum of all the values of any algebraical function of those lines.

Most of the properties of circles given by Archimedes are extended to conic sections, and some of the algebraical and geometrical properties of Pappus are rendered more general; and the principles invented applied to many other cases. In the first edition of this book published in 1762 were nearly enumerated the lines of the fourth order on the same principles as Newton's enumeration of lines of the third order; but this has since been rejected by the author as not sufficiently distinguishing the curve, and as being of no great utility.

The first chapter treats of finding the fluxion of a fluent, when the quantity or fluent is considered as generated by motion; or the parts from the whole when the whole or quantity is considered as consisting of innumerable parts. It further gives the law of a series, which expresses the fluxion of an exponential of any order.

Chapter 2, is versant about the fluents of fluxions. 1. It finds the general fluent of a fluxion Px^., when P is any algebraical function of x however irrational but not exponential; for which intent it investigates the common divisors of any two quantities contained under the different vincula; and thence the common divisors of the resulting divisors, and so on; and likewise all the equal divisors contained in any of the abovementioned quantities; whence it so reduces the quantity P, that no equal nor common divisors may be contained in any of the resulting quantities under the different vincula; and from the common method deduces the terms of a series to the number, which the series is shewn to consist of, when it does not proceed in infinitum. 2. It demonstrates, that if the dimensions of x in the denominator of P exceed its dimensions in the numerator by 1, then the fluent cannot be expressed in finite terms; and also if one factor of P be (A ± (A² + a)^1/2)^l, where a is an invariable quantity, and in some other cases the substitution required must be somewhat different. 3. The fluents of some fluential and exponential fluxions, or fluxions involving fluents and exponential quantities, are given. 4. A general method of discovering whether the fluent of any fluxion of any order involving one, two or more variable quantities, and their fluxions, can be expressed in terms of the variable quantities and their fluxions. 5. The correction of fluents of all orders, and thence the fluent contained between any values of the variable quantities and their fluxions, is given; in these corrections the same roots of the irrational quantities are to be used in the correction as in the fluent. 6. From the transformation of equations and the principles before delivered, are deduced fluents equal to each other. 7. Some exponential quantities given which continually change from possibility to impossibility, and from impossibility to possibility. 8. Is a method of finding whether the fluent of any fluxion contained between any limits are finite or not. 9. The sum of the fluents of a fluxion which is. an algebraical function of the letter x multiplied into x^. can always be expressed by finite terms, circular arcs and logarithms, the extraction of the roots of equations being granted. | 10. Some fluxions involving irrational quantities are reduced to others, in which no irrationality is contained. 11. The general principles of deducing whether the fluent of a given fluxion can generally be expressed by finite algebraical terms, their circular arcs and logarithms. 12. Some equal correspondent fluents are found by substitutions deduced from equations in which two variable quantities are similarly involved. 13. Some necessary corrections are given of finding the fluents of all the fluxions of the formula x^{pn ± sn - 1} x^. X R^{m ± l} X S^{o ± m} X T^{t X n} X &c, (where s, l, m, n, &c denote any whole numbers, and from a + b + g + &c, independent fluents; but perhaps not from a + b + g + &c fluents, which have different values of the quantities, s, l, m, n, &c. 14. The number of independent fluents of the formulæ x^{q + an + bm} X (a + bxⁿ + cx^m)^{l + p} X x^., where a, b and p denote whole affirmative numbers, &c; and the number of independent fluents of the formulæ X^.∫Yx^., where X^. is a fluxion of which the fluent can be sound, from which can be deduced all of the same formula, is immediately known from the number of independent fluents of the formula Yx^. and XYx^. which determine all of those formulæ. 15. Let , and from some fluents of the fluxions of the formulæ p X x^{mn - 1} x^., where m is a whole affirmative number, are determined the remaining ones of the same formula. 16. Something is added concerning finding the value of a fraction, when both the numerator and denominator vanish; and lastly from the fluents of some fluxions being given, the method of deducing the fluents of others.

Chapter 3, principally treats of algebraical and fluxional equations. 1. It gives the method of transforming two or more fluxional equations into one so as to exterminate one or more variable quantities and their fluxions, and finds the order of the resulting equation. 2. It reduces some fluxional equations into more. 3. A method of reducing fluxional equations involving fluents so as to exterminate the fluents. 3. Some cases are given, in which the two variable quantities contained in a given equation are expressed in terms of a third. 4. Given an algebraical equation expressing the relation between x and y; a method is given of finding the fluent of yx^.n or other fluxions in finite terms of x and y, if they can be expressed by such; or else by infinite series; this was first taught in the Philosophical Transactions in the year 1764. 5. Something is added concerning the correction of fluxional equations. 6. A method of investigating, whether a given equation is the general fluent of a given fluxional equation. 7. The method of deducing, whether a given equation is a particular or general fluent of a given fluxional equation. In both by substituting for the fluxions their values deduced from the fluential equation their values &c in the fluxional, the fluxional must result = 0; and in the general fluent there must be contained so many invariable quantities to be assumed at will independently as is the order of the fluent; and in both all the variable quantities must necessarily be variable, and no function of them vanish out of the fluxional equation from the substitution; for then all the conditions of the fluxional equation are answered by the fluential. 8. An investigation, when fluxional equations are integrable. 9. From some fluents are deduced others, e. g. if the area between any two ordinates to one abscissa can in general be found, then the area between any two ordinates of any other abscissa can be found &c. 10. From given fluxional equations and the fluents of some fluxions are deduced the fluents of many others. 11. The fluent of the first order of a fluxional equation of the nth order will have (n) different values and n different multipliers; and the fluent of the second order n.(n - 1)/2 different values, &c. 12. Let a = 0, b = 0, g = 0, &c, (n) general fluents of the fluxional equation, l = 0, then will any function of the fluents a, b, g, &c be a fluent of the same fluxional equation l = 0. 13. From assuming equations, which contain only simple powers of the invariable quantities to be assumed at will, may easily be deduced fluxional equations, of which the general resolutions are known: 2. From assuming the values of any variable quantities and substituting then their fluxions for the variable quantities, &c. in any functions p, r, &c of the variables assumed, let the quantities resulting be A, B, &c; then generally will p = A, r = B, &c. be fluxional equations, of which the particular fluentials are known. It may be observed in this place as before, that from no general reasoning can particular fluents be deduced. 14. In the resolution of fluxional equations it is observed, that from the logarithmic and exponential quantities contained in the fluxional, may be deduced by chapter 1 the exponentials &c contained in the fluential: 2, and in a similar manner from the irrational quantities and denominators contained in it, the correspondent irrational quantities and denominators contained in the fluential: 3, the greatest dimensions of y multiplied into x^. must be greater than those of y into y^. by unity; when there are two of this kind &c, (dx^. + ey^.) the refolution is given; and so of more. 15. In the given equation, if the fluxion of the greatest order does not ascend to one dimension only; then by extraction &c so reduce the equation, that it may ascend to one dimension only; and thence find the fluent of any fluxion Pⁿy^. + Q^{n - 1}y^. + &c, + R′ⁿz^. + &c. 16. Let a fluxional equation be given involving x and y, in which x flows uniformly, a method is given of finding whether it admits of a multiplier, which is a function of x ∴ and similarly of multipliers of other formulæ. 17. The method of deducing the multipliers of fluxional equations by infinite series. 18. Some fluxional equations are reduced by substitutions, which substitutions are commonly easily deducible from the fluxional equation given. 19. Somewhat concerning the reduction of some fluxional equations to homogeneous, and concerning homogeneous equations of different orders; and of reducing an homogeneous fluxional equation of n order to a fluxional equation of n - 1 order: and | also of reducing m fluxional equations of n order to one of mn - 1 orders, and so of all others to one degree less than the order generally occurring if they had not been homogeneous. 20. The substitution of an exponential for a variable quantity in equations which contain no exponential quantity; for sometimes n has been substituted for a quantity which flows uniformly, and then w supposed to flow uniformly, which leads to a false resolution. 21. A caution is given not to substitute homogeneous functions of no dimensions for variable quantities; and in the general resolution to observe, that there is contained an invariable quantity to be assumed at will, which is not contained in the fluxional equation. 22. Something more added concerning the fluents of , where p, q, r, &c. are functions of x, and so of some other fluxional equations. 23. Fluxional equations are deduced, of which the variable quantities cannot be expressed in terms of each other, but both may be expressed in terms of a third. 24. Every fluxion or fluent which is a function x, y, z, and x, y. &c. is expressed in terms of partial differences. 25. The resolution of some equations expressing the relation between partial differences &c is given. 26. Some observations on finding the fluents of fluxions, when the variable quantities become infinite.

The second book treats of increments and their integrals. 1. Some new laws of the increments are given. 2. The fluxion of the increment of P will be equal to the increment of the fluxion; where P is any function of x, if only the fluxion of the increment of x be equal to the increment of the fluxion. 3. Increments are reduced to others of given formulæ e. g. a + b/x + g/(x(x+x_.)) +, &c, and it is observed that if b be not = 0, then the integral cannot be found in finite terms of the variable quantity, &c. It may be observed, that Taylor, Monmort, &c, first found the integral of the two increments x.―(x-x_.).―(x-2x_.) ... ―(x-―(n-1) x_.) and 1/(x.―(x-x_.) ...―(x-―(n-1)x_.)) but did not proceed much further (correspondent to the finding the fluxion of the fluent xⁿ); the increments of fluents have been since deduced, &c. In this book are discovered propositions correspondent to most of the inventions in fluxions, e. g. a method of finding the integral of any increment expressed in algebraical or exponential terms of the variable quantity or quantities, and when the fluent cannot be expressed: it is observed that they cannot be expressed in finite terms of the variable x, &c, if the dimensions of x, &c, in the denominator exceed its dimensions in the numerator by 1; or if any factor in the denominator of the fraction reduced to its lower terms have not another contained likewise in the denominator, distant by a whole number, multiplied into the increment of x. —The increments of some integrals are deduced from the integrals of other increments; the integrals of some incremental equations from different methods; their general integrals, and particular corrections, &c, &c; but here it is to be observed, that the general problem of increments cannot be extended beyond the particular of fluxions, but somewhat more may be added, when both are joined together. The third book is versant concerning infinite series. 1. It gives the ratio of the apparent and real convergency. 2. A method of finding limits between which the sum of the series consists; and also whether the sum of the series is sinite or not from the terms being given or equation between the terms. 3. The convergency of the whole series is judged from the ratio of convergency of the terms at an insinite distance. 4. The series from the fluent converges, if the series from the fluxion does, there are several propositions on insinite series deducible from the common algebra. 5. Let an equation ; and b/a much greater than c/b, c/b than d/c; &c. then will all the roots be possible, and a/b an approximation to the least root, b/c to the next, &c: if an equation , and if one root be much less than any m root, but much greater than the remaining; or if the equation be , then will the approximation to the above root be i/h - (k/i - (gi²)/h³) + &c. 6. Somewhat on the approximations when the approximation given is much more near to one, two, or more roots than to any other, and on the degree of convergency of the subsequent approximations deduced; and their ultimate approximations. 7. Given approximations to m roots of a given equation are deduced more near approximations to them. 8. The incremental equation given and applied to approximations. 9. From given approximations to two or more unknown quantities contained in two or more equations are deduced more near approximations to them, either when the approximations given are more near to one, or to two, or more roots of one or more of the unknown quantities than to any others, and so of infinite equations. 10. New series are given for the fluents of different fluxions. 1. . The sine of the arc A±e is S ± Ce - 1/2 Se², &c, and cosine of the same arc = C± Se - 1/(2.3) Ce² ± &c. S and C being the sine and cosine of A, the fluent of the fluxion of an elliptical arc √((1-cx²)x^.)/√(1 - x³) which differs little from the arc of a circle when e is a very small quantity = A′ - c/2 X (1.A - xP)/2 - &c, where , | , and A = arc of a circle of which the sine is x.

A similar series may be applied from the arc of an hyperbola or ellipse, to find a correspondent arc of an hyperbola or ellipse not much different from the preceding. In this method the series proceeds according to the dimensions of some small quantities, and the first term of the series is generally a near value of the quantity sought. These series properly instituted will generally converge the swiftest. 11. Something new is added concerning the fluent of the fluxional equation ; E and F being any quantities to be assumed at will; and of correspondent equations to logarithms, and finding their values when z is increased by e. 12. A series for the increase of the arc from a small increase of the tangent, fine, &c. 12. When the terms a and x of the binomial a±x are equal, the cases are given in which the series or the series a^mx = (m/2)a^m-1 x + &c, &c. will ultimately converge. 13. If any algebraical quantity V a function of x be reduced into a series proceeding according to the dimensions of x, a general method of finding what are the limits between which it converges; or the series from ∫ Vx^., &c; and the method of interpolations so as to render them converging. 14. The convergency of different series are compared together. e. g. is given : there is an erratum contained in this example, for a - is sometimes printed instead of a +: this series is easily deduced from Bernouilli's method of deducing infinite series, and has been since printed in the Philosophical Transactions. 15. Given algebraical or fluxional equations, and a fluxional quantity, a method is given of finding a series, which expresses the fluent of the fluxional quantity, from which principles are deduced new series for the area of a segment of a circle, the periphery of the ellipse, hyperbola, &c. 16. It is shewn, that serieses proceeding according to the dimensions of a quantity x always diverge, when serieses for the same purpose proceeding according to the reciprocal of its dimensions converge; unless sometimes in the case when they both become the same. 17. As series proceeding in infinitum according to the dimensions of the quantity x were first invented or used for the finding the fluents of fluxions, it being reduced into terms, whose fluents were known: so in finding integrals of increments it may be necessary to reduce the quantity into an infinite series of terms, whose integrals are known, and which converges. Examples of formulæ of serieses of this kind are given. 18. Methods are given of finding the value of one unknown quantity contained in one or more equations involving more unknown quantities, and the law of their convergencies and the interpolations necessary to render serieses for finding fluents converging, similar principles may be applied to incremental and fluxional equations. 19. It is observed, that in finding the value of any variable quantity in a series proceeding according to the dimensions of another, there will occur in a fluxional or incremental equation of (n) order in the series n invariable quantities to be assumed at will; and also the fluxional equations, &c. from whence they will arise. 20. The finding the integral of ż/z, &c. 21. From the correspondent relation between the sums of two series resulting, which are functions of a variable quantity y, when the relation between x and z two values of y are given, is given a method of finding the coefficients of the series. 22. The rule generally called the reductio ad absurdum extended to more substitutions.

The fourth book treats of the summation of series, a method of correspondent values and several other problems. 1. Of finding the sum of a series expressed by a rational function of z into xⁿ²; where z denotes successively the numbers 1, 2, 3, &c, in infinitum. 2. Given an equation expressing the relation between the successive sums, the relation between the successive terms is known, and the vice versa, &c. 3. It is found from an equation expressing the relation between the successive sums, terms and z the distance from the first term of the series, whether the sum of the series is finite or not. 4. The difference between z^-0 and ―(z+1)_-0, where z denotes the distance from the first term of the series, will be — 0 X z^-0-1, which is greater than the simple ratio let 0 be as small as possible, and consequently the sum of the series finite. 5. If a series a + bx + cx² + x³, of which at an infinite distance the preceding coefficients have to the subsequent the ratio of r:1, be multiplied into a function = 0, when x=d, then if a be greater than r the series will diverge; if less converge. 6. From adding several terms of one or more series together may be formed a series, of which the sum from the sums of the preceding series is known. 6. Serieses are formed, of which the sums are known from varying the divisors, &c. 7. From given series are deduced others, of which the sums are known, and the sum of many series are deduced from finding the fluxions of fluents and fluents of fluxions. 8. From the relation between the different terms given is deduced the correspondent fluxional equation. 9. The finding the terms of any series, which can be deduced from given series; and thence deducing many series of which the sums can be found from the sum of the given series. 10. Series are given of which the sums can be found from finite terms, circular arcs, logarithms, elliptical and hyperbolical arcs. 11. From a general expression, when algebraical, fluxional, incremental, &c, for the sum of a series can be deduced a similar expression for the sum of every second, third, &c, terms. 12. An infinite series may be a particular resolution of infinite fluxional equations. 13. The terms of some series may be infinite and their sums known. 14. The general fluent of is given by a series of the same kind, and the same of some other fluxional equations. 15. A quantity is found which multiplied into a series | more swiftly converging gives a given series. 16. The first differences of the terms of some series are given; if the terms are in geometrical ratio to each other the abovementioned differences will also be in geometrical ratio to each other: whence it appears, that the series from this method of differences will converge least when the given series converges swiftest, &c, but not always the contrary. Several other propositions are added concerning the method of differences applied to series. 17. A parabolico-hyperbolical curve is drawn through any number of points, as also an algebraical solid —. 18. Something is given concerning the convergency &c. of series deduced from the differences of the numerators of a given series, of which the denominators constitute a geometrical progression. 19. A rule is given for rendering series converging, in which it is observed that the sum of so many terms should be found that z the distance from the first term of the series may exceed the greatest root of the equation resulting from the quantity which expresses the term made = 0. 20. An equation expressing the relation between the sums and terms is reduced to an infinite fluxional equation expressing the relation between the sum or term, its fluxions, and z the distance from the first term of the series. 21. From a method being known of finding the sum of a series, which involves one variable only, is given a method of finding the sum of series which involve more variable quantities: and from assuming sums of serieses of this kind are deduced their terms. 22. The sums of series are found consisting of irrational terms. 23. The principle of the convergency of the approximations found in drawing parabolical curves through given points. 24. Something new is given concerning the interpolations of quantities. 25. . if a, b, g, &c, are the roots of , &c. 26. Something is added concerning series from . 27. Nandens's Problems are somewhat extended. 28. Something is added on changing continual fractions into others. 29. A method of transforming series into continual factors. 30. A rule for finding the sine and cosine of n/m the arc; and transforming an algebraical equation into an equation expressed in terms of sines and cosines, and thence from an approximation to the sine is found one more near; the same might have been performed by tangents, cotangents, secants, cosecants, &c. 31. From some-fluents given have been found others, and consequently by reducing the fluents to infinite series from some infinite series given may others be deduced. 32. The fluent of (xa x^.)/(1 ± xⁿ) is found by approximation, where a is an irrational quantity, which method of finding approximations to the indices may be applied to other cases. 33. The sum of the fractions are found when the denominators = 0, and consequently each particular in- finite. 34. It is asserted, that the sum of certain fractions given become = 0, when the terms are expressed by a fraction of which the denominator is a rational function of the distance from the first term of the series. 35. ∫x^{a - b - 1}x^.∫x^{b - g - 1}x^. ∫x^{g - d - 1}x^. X P, where , &c, will be to ∫x^{b - a - 1}x^. ∫x^{a - g - 1}x∫x^{g - d - 1}x^.∫&c. X P :: x^z : x^b if the fluents are contained between the same values of x. 36. Are given some series consisting of two, of which the one converges, when the other diverges, and consequently the sum of both diverges; &c. 37. From the law of a series being given, the law of the series which expresses the square, or some function of the given series, is found.

1. A method of differences, which deduces from the sums given any successive sums, e. g. Let S¹, S², S³, S⁴, be the logarithms of the ratios r : r + p, r : r + 2p, r : r + 3p, r : r + 4p, then will the logarithm of r : r + 5p be 5 X (S⁴ - S²) + 10(S² - S³) nearly: then rules are given in general, and likewise their errors from the true values.

2. A method of correspondent values is given, e. g. Let a, b, c, d, &c, be values of x; and S^a, S^b, S^c, S^d, &c, correspondent values of y; then may .

3. If the formula of the series be ; if the formula of the series be , which answers to Briggs's or Newton's method of interpolations; or the series will be x^h/a^h X ((x^k - b^k) (x^k - c^k) (x^k - d^k) &c)/((a^k - b^k) (a^k - c^k) (a^k - d^k) &c) X S^a + x^h/b^h X ((x^k - a^k) (x^k - c^k) (x^k - d^k) &c)/((b^k - a^k) (b^k - c^k) (b^k - d^k) &c) X S^b + &c; if the formula of the series be a general formula, which includes the preceding.

5. The series is given for deducing others when the number of correspondent values given are either even or odd, and the values of x are equidistant from each other. 6. And also from correspondent values of x and y to a number of equidistant values of x is deduced the value of y to the next successive or any successive value of x. 7. Some arithmetical theorems are deduced from the preceding propositions. 8. Another method is given of resolving the preceding problem. 9. A method of correcting the solution from a solution | given which finds (n) values of y to (n) given values of x true, and m false to (m) other values. 10. A similar resolution is added from correspondent values of x, y, z, &c given; and more general resolutions. 11. Given the resolution of some cases, and formula in which the general is contained, a method is given in some cases of deducing it. 12. The principles of a method of deductions and reductions are added.

In a Pamphlet published at Cambridge, algebraical quantities are translated into probable relations, and some theorems on probabilities thence deduced; to which are adjoined,

1. The theorem ; this becomes the binomial theorem when l = 0; and it will afford answers to similar cases when the whole number of chances are increased or diminished constantly by l, as the binomial does when they remain the same, a similar multinomial theorem is given. In the same pamphlet are further added some new propositions on chances, on the values of lives, survivorships, &c. In these books are also contained the inventions of others on similar subjects, which in the prefaces are ascribed to their respective authors.

In the Philosophical Transactions are given some properties of numbers, &c, of which some have been published in the books above mentioned; to which may be subjoined something in mixed mathematics, viz, a paper on central forces, which extends not only to central forces, but also to forces applied in any other direction, as in the direction of the tangent, and consequently includes resistances, &c. It gives a rule for finding the forces tending to two or more given points when the curve described and velocity of the body in every point of it is given, e. g. Let the curve be an ellipse, and the velocity the same at every point, and the two centres of force be the foci of the ellipse; then will the forces tending to the two foci be equal, and vary as the square of the sine of the angle contained between the distance from the centre of force to the point in which the body is situated, and the tangent to the curve at that point.

The method of deducing the fluxional equations which express the curve described by a body acted on by any forces tending to given points, or applied in any given directions: some other propositions are contained on similar subjects. 2. A paper on the fluxions of the attractions of lines, surfaces, and solids, and from the different methods of deducing them are found different fluents equal to each other: a third paper gives a solution of Kepler's problem of cutting the area of a circle described round a point by approximations, which also is applied to other cases; this like- wise contains some other problems. Many of these discoveries have since been published, some in the London, and other foreign transactions.

Let , then will l denote the log. of N to the modulus e. If e the modulus = 10, then will the system be the common or Briggs's system of logarithms. Logarithms, and the sums of some other serieses, of the formulæ ax^h + bx^{h + k} + &c may be deduced in a manner similar to that which was used by the Ancients for finding the sines of the arcs of circles.

To particularise the numerous propositions contained in these works, would exceed the limits of our design. Besides those already mentioned, others are interspersed through the whole works.