FUNCTION  . 12 . The processes of the integral calculus consist largely in trans-formations of the functions to be integrated into such fndef.+nite forms that they can be recognized as See also:

• DIFFERENTIAL

differential co- efficients lnteara/s. of functions which have previously been differ- entiated . Corresponding to the results in the table of § II we have those in the following table: f(x) ff(x)dx --- xa+u x° n+I for all values of n except - I x See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

log rx See also:

• EA(nar...), (p.s„,.,.)
• EA(psr...), (p, „...)• (pqr...)

ea: a-See also:

• LEA, HENRY CHARLES (1825–1909)

lea: cosx See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

sin x sin x -See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

cos x (¢2-x2)-z x sin la a tan ' x a'2+x2 The formal rules of § 11 give us means for the transformation of integrals into recognizable forms . For example, the See also:

• RULE

rule (ii.) for a sum leads to the result that the integral of a sum of a finite number of terms is the sum of the integrals of the several terms . The rule (iii.) for a product leads to the method of integration by parts . The rule (v.) for a function of a function leads to the method of substitution (see 6 d8 hPlnw) either Rules of Differentiation . y a II . See also:

• HISTORY

History . 13 . The new limiting processes which were introduced in the development of the higher See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

analysis were in the first instance See also:

• KEPLER, JOHANN (1571-1630)

Kepler's related to problems of the integral calculus . Johannes methods Kepler in his Astronomia nova .

. . de motibus stellae of Integra- Martis (16og) stated his See also:

• LAWS

laws of planetary See also:

• MOTION (Lat. motio, from movere, to move)
• MOTION, LAWS OF

motion, to See also:

• LION
• LION (Lat, leo, leonis; Gr. Mew)

lion. the effect that the orbits of the See also:

• PLANETS, MINOR

planets are ellipses with the See also:

• SUN
• SUN (0. Eng. sonne, Ger. sonne. Fr. soleil, Lat. sol, Gr. ijXior, from which comes helio- in various English compounds)

sun at a See also:

• FOCUS (Latin for " hearth " or " fireplace ")

focus, and that the radii vectores See also:

• DRAWN

drawn from the sun to the planets describe equal areas in equal times . From these statements it is to be concluded that Kepler could measure the areas of See also:

• FOCAL

focal sectors of an See also:

• ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

ellipse . When he made out these laws there was no method of evaluating areas except the See also:

• GREEK
• GREEK, ETRUSCAN AND

Greek methods . These methods would have sufficed for the purpose, but Kepler invented his own method . He regarded the See also:

• AREA

area as measured by the " sum of the radii " drawn from the focus, and he verified his laws of planetary motion by actually measuring a large number of radii of the See also:

• ORBIT (from Lat. orbita, a track, orbis, a wheel)

orbit, spaced according to a rule, and adding their lengths . He had observed that the focal See also:

• RADIUS

radius vector SP (fig . 5) is equal to the perpendicular SZ drawn from S to the tangent at p to the See also:

• AUXILIARY (from Lat. auxilium, help)

auxiliary circle, and he had further established the theorem which we should now See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express in the See also:

• FORM (Lat. forma)

form—the differential See also:

• ELEMENT (Lat. elementum)

element of the area See also:

• ASP ( Vipera aspis)

ASp as Sp turns about S, is equal to the product of SZ and the differential add), where a is the radius of the auxiliary circle, and 4 is the See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle ACp, that is A the See also:

• ECCENTRIC (from Gr. F,c, out of, and Kivrpov, centre)

eccentric angle of P on the ellipse . The area ASP bears to the area ASp the ratio of the See also:

• MINOR
• MINOR (Lat. for smaller, lesser)
• MINOR, ROBERT CRANNELL (1839-1904)

minor to the See also:

• MAJOR
• MAJOR (Lat. for " greater ")
• MAJOR (or MAIR), JOHN (1470-1550)

major See also:

• AXIS (Lat. for " axle ")

axis, a result known to according to the rule that the eccentric angles of their ends are equidifferent, and his " sum of radii " is proportional to the expression which we should now write f 0(a+ae cos 4 )d4 , where e is the eccentricity . Kepler evaluated the sum as proportional to d)+e sin o . Kepler soon afterwards occupied himself with the volumes of solids . The vintage of the See also:

• YEAR

year 'fors was extraordinarily abundant, and the question of the cubic content of See also:

• WINE (Lat. vinum, Gr. oivos)

wine casks was brought under his See also:

• NOTICE

notice . This fact accounts for the See also:

• TITLE (0. Fr. title, mod. titre, from Lat. titulus)

title of his See also:

• WORK

work, Nova stereometria doliorum; accessit stereometriae Archimedeae supplementum (1615) .

In this See also:

• TREATISE

treatise he regarded solid bodies as being made up, as it were (veluti), of " infinitely " many " infinitely " small cones or " infinitely " thin disks, and he used the notion of summing the areas of the disks in the way he had previously used the notion of summing the focal radii of an ellipse . 14 . In connexion with the See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

early history of the calculus it must not be forgotten that the method by which logarithms were invented (1614) was effectively a method of infinitesimals . Natural logarithms were not invented as the indices of a certain See also:

• BASE

base, and the notation e for the base was first introduced by See also:

• EULER, LEONHARD (1707-1783)

Euler more than a See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century after the invention . Logarithms were introduced as See also:

• NUMBERS, BOOK OF
• NUMBERS, PARTITION OF

numbers which increase in See also:

• ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

arithmetic progression when other related numbers increase in geometric progression . The two sets of numbers were supposed to increase together, one at a See also:

• UNIFORM

uniform See also:

• RATE

rate, the other at a variable rate, and the increments were regarded for purposes of calculation as very small and as accruing discontinuously . 15 . Kepler's methods of integration, for such they must be called, were the origin of See also:

• BONAVENTURA, SAINT (JOHN OF FIDANZA)

Bonaventura Cavalieri's theory of Cava- the summation of indivisibles . The notion of a See also:

• LIEN

lien's continuum, such as the area within a closed See also:

• CURVE (Lat. curvus, bent)

curve, Inds- es. as being made up of indivisible parts, " atoms " of visib area, if the expression may be allowed, is traceable to the speculations of early Greek philosophers; and although the nature of continuity was better understood by See also:

• ARISTOTLE
• ARISTOTLE (384-322 B.C.)

Aristotle and many other See also:

• ANCIENT
• ANCIENT (also spelt ANTIENT; derived, through the Fr. ancien, old, from the late Lat. antianum, from ante, before)

ancient writers yet the unsound atomic conception was revived in the 13th century and has not yet been finally uprooted . It is possible to contend that Cavalieri did not himself hold the unsound See also:

• DOCTRINE

doctrine, but his See also:

• WRITING
• WRITING (the verbal noun of " to write," O. Eng. writan, to inscribe)

writing on this point is rather obscure . In his treatise Geometria indivisibilibus continuorum nova quadam ratione promota (1635) he regardeda See also:

• PLANE

plane figure as generated by a See also:

• LINE

line moving so as to be always parallel to a fixed line, and a solid figure as generated by a plane moving so as to be always parallel to a fixed plane; and he compared the areas of two plane figures, or the volumes of two solids, by determining the ratios of the sums of all the indivisibles of which they are supposed to be made up, these indivisibles being segments of parallel lines equally spaced in the See also:

• CASE
• CASE, JOHN (d. 1600)

case of plane figures, and areas marked out upon parallel planes equally spaced in the case of solids . By this method Cavalieri was able to effect numerous integrations See also:

• RELATING

relating to the areas of portions of conic sections and the volumes generated by the revolution of these portions about various axes .

At a later date, and partly in See also:

• ANSWER (derived from and, against, and the same root as swear)

answer to an attack made upon him by See also:

• PAUL
• PAUL (PAULUS)

Paul Guldin, Cavalieri published a treatise entitled Exercitationes geometricae See also:

• SEX (Lat. sexes; possibly connected with secure, to cut)

sex (1647), in which he adapted his method to the determination of centres of gravity, in particular for solids of variable See also:

• DENSITY (Lat. densus, thick)

density . Among the results which he obtained is that which we should now write Jy xm+I ax'"dx m+I,(m integral) . He regarded the problem thus solved as that of determining the sum of the mth See also:

• POWERS, HIRAM (1805-1873)

powers of all the lines drawn across a parallelogram parallel to one of its sides . At this See also:

• PERIOD
• PERIOD (Gr. irepioSos, a going or way round, circuit, aepi, round, and &Sis, way, road)

period scientific investigators communicated their results to one another through one or more intermediate persons . Such intermediaries were See also:

• PIERRE

Pierre de Carcavy and See also:

• PATER, WALTER HORATIO (1839-1894)

Pater Marin See also:

• MERSENNE, MARIN

Mersenne; and among the writers thus in communication were Bonaventura Cavaliers, Christiaan See also:

• HUYGENS, CHRISTIAAN (1629-1695)
• HUYGENS, SIR CONSTANTIJN (1596-1687)

Huygens, Galileo Galilei, See also:

• GILES (GIL, GILLES), ST

Giles Personnier de See also:

• ROBERVAL, GILLES PERSONNE (or PERSONIER) DE (1602-1675)

Roberval, Pierre de See also:

• FERMAT, PIERRE DE (1601-1665)

Fermat, Evangelista See also:

• TORRICELLI, EVANGELISTA (1608—1647)

Torricelli, and a little later Blaise See also:

• PASCAL, BLAISE (1623-1662)
• PASCAL, JACQUELINE (1625-T661)

Pascal; but the letters of Carcavy or Mersenne would probably come into the hands of any See also:

• MAN
• MAN, ISLE OF (anc. Mona)

man who was likely, to be interested in the matters discussed . It often happened, that, when some new method was invented, or some new result obtained, the method or result was quickly known to a wide circle, although it might not be printed until after the See also:

• LAPSE (Lat. lapses, a slip or departure)

lapse of a See also:

• LONG, GEORGE (1800-1879)
• LONG, JOHN DAVIS (1838– )

long See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time . When Cavalieri was See also:

• PRINTING (from Lat. imprimere, O. Fr. empreindre)

printing his two See also:

• TREATISES

treatises there was much discussion of the problem of quadratures . Roberval (1634) regarded an area as made up of " infinitely " many " infinitely " narrow strips, each of which may be considered to be a rectangle, and he had similar ideas in regard to lengths and volumes . He knew how to approximate to the quantity which we express by f px'"dx by the See also:

• PROCESS

process of forming the sum 0 1 I" }2 { ... (n—1)'" nm+1 and he claimed to be able to prove that this sum tends to 1((m+1), as n increases for all See also:

• POSITIVE (or PORTABLE) ORGAN

positive integral values of m . The method of integrating x°" by forming this sum was found also Fermat's by Fermat (1636), who stated expressly that he method of arrived at it by generalizing a method employed by Integra-See also:

• DON
• DON (anc. Tanais)

don . See also:

• ARCHIMEDES (c. 287–212 B.C.)
• ARCHIMEDES, SCREW OF

Archimedes (for the cases m=1 and m= 2) in his books on Conoids and Spheroids and on Spirals (see T .

L . See also:

• HEATH
• HEATH, BENJAMIN (1704-1766)
• HEATH, WILLIAM (1737—1814)

Heath, The See also:

• WORKS

Works of Archimedes, See also:

• CAMBRIDGE
• CAMBRIDGE, EARLS AND DUKES OF
• CAMBRIDGE, RICHARD OWEN (1717-1802)

Cambridge, 1897) . Fermat extended the result to the case where m is fractional (1644), and to the case where m is negative . This latte' See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

extension and the proofs were given in his memoir, Proportions geometricae in quadrandis parabolis et hyperbolis usus, which appears to have received a final form before 1659, although not published until 1679 . Fermat did not use fractional or negative indices, but he regarded his problems as the quadratures of parabolas and hyperbolas of various orders . His method was to See also:

• DIVIDE

divide the See also:

• INTERVAL

interval of integration into parts by means of intermediate points the Abscissae of which are in geometric progression .. In the process of § 5 above, the points M must be chosen according to this rule . This restrictive See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition being understood, we may say that Fermat's formulation of the problem of quadratures is the same as our See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition of a definite integral . The result that the problem of quadratures could be solved for any curve whose See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation could be expressed in the form y=x"'(m'p—I), or in the form y = alx'"l+a2x'"2+ ... +a"x", L ogarithms . Successors of Cavalieri . where none of the indices is equal to -1, was used by See also:

• JOHN
• JOHN (1167–1216)
• JOHN (1290-c. 1320)
• JOHN (1296-1346)
• JOHN (1371–1419)
• JOHN (1468-1532)
• JOHN (1801-1873)
• JOHN (Heb. llni')
• JOHN (ZAPOLYA) (1487-1540)
• JOHN, 4TH MARQUESS OF TWEEDDALE (c. 1695-1762)
• JOHN, DON (1545-1578)
• JOHN, DON (1629–1679)
• JOHN, GOSPEL OF ST
• JOHN, or HAYS (1513–1571)
• JOHN, THE APOSTLE
• JOHN, THE EPISTLES OF

John See also:

• WALLIS, JOHN (1616-1703)

Wallis in his Arithmetica infinitorum (1655) as well as by Fermat (1659) .

The case in which m=—1 was that of the various See also:

• ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary rectangular See also:

• HYPERBOLA

hyperbola; and See also:

• GREGORY
• GREGORY (Gregorius)
• GREGORY (Grigorii) GRIGORIEVICH ORLOV, COUNT (1734-1783)
• GREGORY, EDWARD JOHN (1850-19o9)
• GREGORY, OLINTHUS GILBERT (1774—1841)
• GREGORY, ST (c. 213-C. 270)
• GREGORY, ST, OF NAZIANZUS (329–389)
• GREGORY, ST, OF NYSSA (c.331—c. 396)
• GREGORY, ST, OF TOURS (538-594)

Gregory of Integra- lions . St See also:

• VINCENT (or VINCENTIUS), ST
• VINCENT, GEORGE (1796-1831?)
• VINCENT, MARY ANN (1818-1887)

Vincent in his See also:

• OPUS ('O7rois)

Opus geometricum quadraturae circuli et sectionum coni (1647) had proved by the method of exhaustions that the area contained between the curve, one asymptote, and two ordinates parallel to the other asymptote, increases in arithmetic progression as the distance between the ordinates (the one nearer to the centre being kept fixed) increases in geometric progression . Fermat described his method of integration as a logarithmic method, and thus it is clear that the relation between the See also:

• QUADRATURE (from Lat. quadratura, a making square)

quadrature. of the hyperbola and logarithms was understood although it was not expressed analytically . It was not very long before the relation was used for the calculation of logarithms by Nicolaus See also:

• MERCATOR, GERARDUS [latinized form of GERHARD KREMER] (1512-1594)

Mercator in his Logarithmotechnia (1668) . He began by writing the equation of the curve in the form y= r/(1+x), See also:

• EXPANDED

expanded this expression in powers of x by the method of See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

division, and integrated it See also:

• TERM

term by term in accordance with the well-understood rule for finding the quadrature of a curve given by such an equation as that written at the See also:

• FOOT

foot of p . 325 . By the See also:

• MIDDLE

middle of the 17th century many mathematicians could perform integrations . Very many particular results had Integra- been obtained, and applications of them had been tion before made to the quadrature of the circle and other conic theintegrafsections, and to various problems concerning the Calculus' lengths of curves, the areas they enclose, the volumes and superficial areas of solids, and centres of gravity . A systematic See also:

• ACCOUNT
• ACCOUNT (through O. Fr. acont, Late Lat. comptum, cornputare, to calculate)

account of the methods then in use was given, along with much that was See also:

• ORIGINAL

original on his See also:

• PART

part, by Blaise Pascal in his Lettres de See also:

• AMOS
• AMOS, SHELDON (1835–1886)

Amos Dettonville sur quelques-unes de ses inventions en geometrie (16J9) . 16 . The problem of See also:

• MAXIMA

maxima and minima and the problem of tangents had also by the same time been effectively solved . Fermatas See also:

• ORESME, NICOLAS (c. 1320-1382)

Oresme in the r4th century knew that at a point where methods of the See also:

• ORDINATE

ordinate of a curve is a maximum or a minimum Differen- its variation from point to point of the curve is slowest; nation .

Ind Kepler in the Stereofnetria doliorum remarked that at the places where the ordinate passes from a smaller value to the greatest value and then again to a smaller value, its variation becomes insensible . Fermat in 1629 was in See also:

• POSSESSION
• POSSESSION (Lat. possessio, possidere, to possess)

possession of a method which he then communicated to one Despagnet of See also:

• BORDEAUX

Bordeaux, and which he referred to in a See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter to Roberval of 1636 . He communicated it to Rene See also:

• DESCARTES, RENE (1596-1650)

Descartes early in 1638 on receiving a copy of Descartes's Gecmetrie (1637), and with it he sent to Descartes an account of his methods for solving the problem of tangents and for determining centres of gravity . Fermat's method for maxima and minima is essentially our method . Expressed in a more See also:

• MODERN

modern notation, what he did was to begin by connecting the ordinate y and the See also:

• ABSCISSA (from the Lat. abscissus, cut off)

abscissa x of a point of a curve by an equation which holds at all points of the curve, then to subtract the value of y in terms of x from the value obtained by substituting x+E for x, then to divide the difference by E, to put E=o in the quotient, and to equate the quotient to zero . Thus he differentiated with respect to x and equated the differential coefficient to zero . Fermat's method for solving the problem of tangents may be explained as follows: Let (x, y) be the coordinates of a point P of a curve, (x', y'), those of a neighbouring point P' on the tangent at P, and let MM'=E (fig . 6) . From the similarity of the triangles P'TM', PTM we have y': A—E=y :A, where A denotes the subtangent TM . The point P' being near the curve, we may substitute in the equation of the curve x—E for x and (yA—yE)/A for y . The equation of the curve is approximately satisfied . If it is taken to be satisfied exactly, the result is an equation of the form 0(x, y, A, E) =o, the See also:

• LEFT

left-See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

hand member of which is divisible by E .

Omitting the See also:

• FACTOR (from Lat. facere, to make or do)

factor E, and putting E =o in the remaining factor, we have an equation which gives A . In this problem of tangents also Fermat found the required result by a process See also:

• EQUIVALENT

equivalent to differentiation . Fermat gave several examples of the application of his method;among them was one in which he showed that he could differentiate very complicated irrational functions . For such functions his method was to begin by obtaining a rational equation . In rationalizing equations Fermat, in other writings, used the See also:

• DEVICE

device of introducing new variables, but he did not use this device to simplify the process of differentiation . Some of his results were published by Pierre Herigone in his Supplementum cursus mathematici (1642) . His communication to Descartes was not published in full until after his See also:

• DEATH

death (Fermat, See also:

• OPERA (Italian for " work ")

Opera See also:

• VARIA (mod. Vicovaro)

varia, 1679) . Methods similar to Fermat's were devised by Rene de Sluse (1652) for tangents, and by Johannes Hudde (1658) for maxima and minima . Other methods for the See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of the problem of tangents were devised by Roberval and Torricelli, and published almost simultaneously in 1644 . These methods were founded upon the See also:

• COMPOSITION
• COMPOSITION (Lat. compositio, from componere, to put together)

composition of motions, the theory of which had been taught by Galileo (1638), and, less completely, by Roberval (1636) . Roberval and Torricelli could construct the tangents of many curves, but they did not arrive at Fermat's artifice . This artifice is that which we have noted in §10 as the fundamental artifice of the infinitesimal calculus .

17 . Among the comparatively few mathematicians who before 1665 could perform differentiations was See also:

• ISAAC (Hebrew for " he laughs," on explanatory references to the name, see ABRAHAM)

Isaac See also:

• BARROW
• BARROW (from A.S. beorh, a mount or hillock)
• BARROW, ISAAC (1630-1677)
• BARROW, SIR JOHN (1764-1848)

Barrow . In his See also:

• BOOK

book entitled Lectiones opticae et geometricae, Barrow's written apparently in 1663, 1664, and published in Differ-1669, 167o, he gave a method of tangents like that entiai of Roberval and Torricelli, compounding two velocities Triangle . in the directions of the axes of x and y to obtain a resultant along the tangent to a curve . In an appendix to this book he gave another method which differs from Fermat's in the introduction of a differential equivalent to our dy as well as dx . Two neighbouring ordinates PM and QN of a curve (fig . 7) are regarded as containing an indefinitely small (indefinite parvum) arc, and PR is drawn parallel to the axis of x . T The tangent PT at P is regarded as identical with the secant PQ, and the position of the tangent is determined by the similarity of the triangles PTM, PQR . The increments QR, PR of the ordinate and abscissa are denoted by a and e; and the ratio of a to e is determined by substituting x+e for x and y+a for y in the equation of the curve, rejecting all terms which are of See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order higher than the first in a and e, and omitting the terms which do not contain a or e . This process is equivalent to differentiation . Barrow appears to have invented it himself, but to have put it into his book at See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

Newton's See also:

• REQUEST, LETTERS OF

request . The triangle PQR is some-times called " Barrow's differential triangle." The reciprocal relation between differentiation and integration (§ 6) was first observed explicitly by Barrow in the book cited above .

If the quadrature of a curve y =f(x) is known, so that the area up to the ordinate x is given by F(x), the curve Barrow's y = F(x) can be drawn, and Barrow showed that the theorsion. subtangent of this curve is measured by the ratio of theorem. its ordinate to the ordinate of the original curve . The curve y=F(x) is often called the " See also:

• QUADRATRIX (from Lat. quadrator, squarer)

quadratrix " of the original curve; and the result has been called " Barrow's See also:

• INVERSION (Lat. invertere, to turn about)

inversion-theorem." He did not use it as we do for the determination of quadratures, or indefinite, integrals, but for the solution of problems of the See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind which were then called " inverse problems of tangents." In these problems it was sought to determine a curve from some See also:

• PROPERTY

property of its tangent, e.g. the property that the subtangent is proportional to the square of the abscissa . Such problems are now classed under "differential_ equations." When Barrow wrote, quadratures were See also:

• FAMILIAR (through the Fr. familier, from Lat. familiaris, of or belonging to the familia, family)

familiar and differentiation unfamiliar, just as hyperbolas were trusted while logarithms were See also:

• STRANGE, SIR ROBERT (1721-1792)

strange . The functional notation was not invented till long afterwards (see FUNCTION), and the want of it is See also:

• FELT (cognate with Ger. Filz, Du. vilt, Swed. and Dan. fill; the root is unknown; the word has given Med. Lat. filtrum, " filter ")

felt in See also:

• READING

reading all the See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics of the 17th century . 18 . The See also:

• GREAT

great See also:

• SECRET (Lat. secretum, hidden, concealed)

secret which afterwards came to be called the " infinitesimal calculus " was almost discovered by Fermat, and still more nearly by Barrow . Barrow went farther than Fermat in the theory of differentiation, though not in the practice, for he compared two increments; he went farther in the theory of integration, for he obtained the inversion-theorem . The great See also:

• DISCOVERY

discovery seems to consist partly in the r x or M'M Flo . 6 . M N FIG . 7 . recognition of the fact that differentiation, known to be a useful process, could always be performed, at least for the functions then known, and partly in the recognition of the fact that the inversion-theorem could be applied to problems of quadrature .

By these steps the problem of tangents could be solved once for all, and the operation of integration, as we See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

call it, could be rendered systematic . A further step was necessary in order that the discovery, once made, should become accessible to mathematicians in See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

general; and this step was the introduction of a suitable notation . The definite See also:

• ABANDONMENT (Fr. abandonnement, from abandonner, to abandon, relinquish; abandonner was originally equivalent to mettred banddn, to leave to the jurisdiction, i.e. of another, bandon being from Low Latin bandum, bannum, order, decree, " ban ")

abandonment of the old tentative methods of integration in favour of the method in which this operation is regarded as the inverse of differentiation was especially the work of Isaac Newton; the precise formulation of See also:

• SIMPLE

simple rules for the process of differentiation in each See also:

• SPECIAL

special case, and the introduction of the notation which has proved to be the best, were especially the work of Gottfried Wilhelm See also:

• LEIBNITZ (LEIBNIZ), GOTTFRIED WILHELM

Leibnitz . This statement remains true although Newton invented a systematic notation, and practised differentiation by rules equivalent to those of Leibnitz, before Leibnitz had begun to work upon the subject, and Leibnitz effected integrations by the method of recognizing differential coefficients before he had had any opportunity of becoming acquainted with Newton's methods . 19 . Newton was Barrow's See also:

• PUPIL (Lat. pupillus, orphan, minor, dim. of pupus, boy, allied to puer, from root pm- or peu-, to beget, cf. "pupa," Lat. for " doll," the name given to the stage intervening between the larval and imaginal stages in certain insects)

pupil, and he knew to start with in 1664 all that Barrow knew, and that was practically all that was known about the subject at that time . His Newton's original thinking on the subject See also:

• DATES

dates from the year lavestiga- tions. of the great See also:

• PLAGUE (in Gr. Xotµos; in Lat. pestis, pestilentia)

plague (1665-1666), and it issued in the invention of the " Calculus of Fluxions," the principles and methods of which were See also:

• DEVELOPED

developed by him in three tracts entitled De analysi per aequationes numero terminorum infinitas, Methodus fluxionum et serierum infinitarum, and De quadratura curvarum . None of these was published until long after they were written . The Analysis per aequationes was composed in 1666, but not printed until 1711, when it was published by See also:

• WILLIAM
• WILLIAM (1143-1214)
• WILLIAM (1227-1256)
• WILLIAM (1J33-1584)
• WILLIAM (A.S. Wilhelm, O. Norse Vilhidlmr; O. H. Ger. Willahelm, Willahalm, M. H. Ger. Willehelm, Willehalm, Mod.Ger. Wilhelm; Du. Willem; O. Fr. Villalme, Mod. Fr. Guillaume; from " will," Goth. vilja, and " helm," Goth. hilms, Old Norse hidlmr, meaning
• WILLIAM (c. 1130-C. 1190)
• WILLIAM, 13TH

William See also:

• JONES
• JONES, ALFRED GILPIN (1824-1906)
• JONES, EBENEZER (182o-186o)
• JONES, ERNEST CHARLES (1819-1869)
• JONES, HENRY (1831-1899)
• JONES, HENRY ARTHUR (1851- )
• JONES, INIGO (1573-1651)
• JONES, JOHN (c. 1800-1882)
• JONES, MICHAEL (d. 1649)
• JONES, OWEN (1741-1814)
• JONES, OWEN (1809-1874)
• JONES, RICHARD (179o-1855)
• JONES, SIR ALFRED LEWIS (1845-1909)
• JONES, SIR WILLIAM (1746-1794)
• JONES, THOMAS RUPERT (1819– )
• JONES, WILLIAM (1726-1800)

Jones . The Methodus fluxionum was composed in 1671 but not printed till 1736, nine years after Newton's death, when an See also:

• ENGLISH

English See also:

• TRANSLATION

translation was published by John Colson . In See also:

• HORSLEY, JOHN (c. 1685 – 1732)
• HORSLEY, JOHN CALLCOTT (1817-1903)
• HORSLEY, SAMUEL (1733-1806)
• HORSLEY, WILLIAM (1774–1858)

Horsley's edition of Newton's works it bears the title Geometria analytica . The Quadratura appears to have been composed in 1676, but was first printed in 1704 as an appendix to Newton's Opticks .

20 . The See also:

• TRACT (from Lat. tractare, to treat of a matter, through Provencal tractat and Ital. trattato)

tract De Analysi per aequationes . was sent by Newton to Barrow, who sent it to John See also:

• COLLINS, ANTHONY (1676-1729)
• COLLINS, JOHN CHURTON (1848-1908)
• COLLINS, MORTIMER (1827-1876)
• COLLINS, WILLIAM (1721-1759)
• COLLINS, WILLIAM (1787–1847)
• COLLINS, WILLIAM WILKIE (1824–1889)

Collins with a request that Newton's it might be made known . One way of making it known method of would have been to See also:

• PRINT

print it in the Philosophical Trans-See also:

• SERIES (a Latin word from serere, to join)

Series actions actions of the Royal Society, but this course was not adopted . Collins made a copy of the tract and sent it to See also:

• LORD
• LORD (O. Eng. hldford, i.e. hldfweard, the warder or keeper of bread, hlIf, loaf; the word is not represented in any other Teutonic language)
• LORD, JOHN (1810-1894)

Lord Brouncker, but neither of them brought it before the Royal Society . The tract contains a general See also:

• PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

proof of Barrow's inversion-theorem which is the same in principle as that in § 6 above . In this proof and elsewhere in the tract a notation is introduced for the momentary increment (momentum) of the abscissa or area of a curve; this " moment " is evidently meant to represent a moment of time, the abscissa representing time, and it is effectively the same as our differential element—the thing that Fermat had denoted by E, and Barrow by e, in the case of the abscissa . Newton denoted the moment of the abscissa by o, that of the area z by ov . He used the letter v for the ordinate y, thus suggesting that his curve is a velocity-time graph such as Galileo had used . Newton gave the See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

formula for the area of a curve v=x'"(m -1) in the form z=x"'+1/(m+1) . In the proof he transformed this formula to the form z" =c"x5, where n and p are positive integers, substituted x+o for x and z+ov for z, and expanded by the See also:

• BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

binomial theorem for a positive integral exponent, thus obtaining the relation z"+nz"-lov+ . . . =c^(xp ~pxp 10 F ...), from which he deduced the relation nz"-lv = c^ pxr-1 by an See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite series, using for this purpose the binomial theorem for negative and fractional exponents, that is to say, the expansion of (i +x)" in an infinite series of powers of x .

This theorem he had discovered; but he did not in this tract See also:

• STATE
• STATE, GREAT OFFICERS OF

state it in a general form or give any proof of it . He pointed out, however, how it may be used for the solution of equations by means of infinite series . He observed also that all questions concerning lengths of curves, volumes en-closed by surfaces, and centres of gravity, can be formulated as problems of quadratures, and can thus be solved either in finite terms or by means of infinite series . In the Quadratura (1676) the method of integration which is founded upon the inversion-theorem was carried out systematically . Among other results there given is the quadrature of curves expressed by equations of the form y=x"(a+bx'")P; this has passed into See also:

• TEXT (Lat. textum, woven fabric, from texere, to weave)

text-books under the title " integration of binomial differentials " (see § 49) . Newton announced the result in letters to Collins and See also:

• OLDENBURG

Oldenburg of 1676 . 21 . In the Methodus fluxionum (1671) Newton introduced his characteristic notation . He regarded variable quantities as generated by the motion of a point, or line, or plane, and called Newton's the generated quantity a " fluent " and its rate of genera- method of tion a " fluxion." The fluxion of a fluent x is represented Fluxions. by x, and its moment, or " infinitely " small increment accruing in an " infinitely " See also:

• SHORT, FRANCIS JOB (1857– )

short time, is represented by to . The problems of the calculus are stated to be (i.) to find the velocity at any time when the distance traversed is given; (u.) to find the distance traversed when the velocity is given . The first of these leads to differentiation . In any rational equation containing x and y the expressions x+xo and y+90 are to be substituted for x and y, the resulting equation is to be divided by o, and afterwards o is to be omitted .

In the case of irrational functions, or rational f unctions which are not integral, new variables are introduced in such a way as to make the equations contain rational integral terms only . Thus Newton's rules of differentiation would be in our notation the rules (i.), (ii.), (v.) -of § together with the particular result which we write (Ix"' dx =mx"-1 (m integral) . a result which Newton obtained by expanding (x+io)'" by the binomial theorem . The second problem is the problem of integration, and Newton's method for solving it was the method of series founded upon the particular result which we write f x"dx=m+1 . Newton added applications of his methods to maxima and minima, tangents and curvature . In a letter to Collins of date 1672 Newton stated that he had certain methods, and he described certain results which he had found by using them . These methods and results are those which are to be found in the Methodus fluxionum; but the letter makes no mention of fluxions and fluents or of the characteristic notation . The rule for tangents is said in the letter to be analogous to de Sluse's, but to be applicable to equations that contain irrational terms . 22 . Newton gave the fluxional notation also in the tract De Quadratura curvarum (1676), and he there added to it notation for the higher differential coefficients and for indefinite integrals, as we call them . Just as x, y, z, . . . are fluents of which .x, 9, z, .

. are the fluxions, so z, y, z, . can be treated as fluents of which the fluxions may be denoted by x, 9, 2', . . . In like manner the fluxions of these may be denoted by y, y, z, . and so on . Again x, y, z, . . may be regarded as fluxions of which the fluents maybe denoted by x, y, i and these again as fluxions of other quantities denoted by x, y, z, . . . and so on . No use was made of the notation c, 3, . . . in the course of the tract . The first publication of the fluxional notation was made by Wallis in the second edition of his See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

Algebra (1693) in the form of extracts from communications made to him by Newton in 1692 . In this account of the method the symbols o, , x . . occur, but not the symbols z, x, . Wallis's treatise also contains Newton's formulation of the problems of the calculus in the words Data aequatione fluentes quotcumque quantitates involvente fluxiones invenire et See also:

• VICE

vice versa (" an equation containing any number of fluent quantities being given, to find their fluxions and vice versa ") .

In the Philosophiae naturalis principia mathematica (1687), commonly called the " Principia," the words " fluxion " and " moment " occur in a lemma in the second book; but the notation which is characteristic of the calculus of fluxions is nowhere used . Nature of the discovery called the tesimal Calculus . by omitting the equal terms z" and c"x5 and dividing the remaining terms by o, tacitly putting o =--o after division . This relation is the same as v=x'" . Newton pointed out that, conversely, from the relation v=xm the relation z=x'n+1/(m-F1) follows . He applied his formula to the quadrature of curves whose ordinates can be expressed as the sum of a finite number of terms of the form ax'"; and gave examples of its application to curves in which the ordinate is expressed x,"+1 Publication of the Fluxional Notation . Retarded Publication of the method of Fluxions . 23 . It is difficult to account for the fragmentary manner of publication of the Fluxional Calculus and for the long delays which took See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place . At the time (1671) when Newton composed the Methodus fluxionum he contemplated bringing out an edition of See also:

• GERHARD
• GERHARD, FRIEDRICH WILHELM EDUARD (1795-1867)

Gerhard Kinckhuysen's treatise on algebra and prefixing his tract to this treatise . In the same year his " Theory of See also:

• LIGHT

Light and See also:

• COLOURS, MILITARY

Colours " was published in the Philosophical Transactions, and the opposition which it excited led to the abandonment of which we write " ;f ydx," but within a See also:

• DAY (O. Eng. dreg, Ger. Tag; according to the New English Dictionary, " in no way related to the Lat. dies")
• DAY, JOHN (1574-1640?)
• DAY, THOMAS (1748-1789)

day or two he wrote " f y." He regarded the See also:

• SYMBOL (Gr. o-uµ(3oXov, a sign)

symbol " f " as representing an operation which raises the dimensions of the subject of operation--a line becoming an area by the operation—and he devised his symbol " d " to represent the inverse operation, by which the dimensions are diminished . He observed that, whereas " f " represents " sum," " d" represents " difference." His notation appears to have been practically settled before the end of 1675, for in See also:

• NOVEMBER (Lat. novem, nine)

November he wrote f ydy = by', just as we do now .

25 . In See also:

• JULY

July of 1676 Leibnitz received an answer to his inquiry in regard to Newton's methods in a letter written by Newton to Oldenburg . In this letter Newton gave a general cones. statement of the binomial theorem and many results pondence relating to series . He stated that by means of such ofNewseries he could find areas and lengths of curves, centres ton and of gravity and volumes and surfaces of solids, but, as Leibnitz. this would take too long to describe, he would illustrate it by examples . He gave no proofs . Leibnitz replied in See also:

• AUGUST (originally Sextilis)

August, stating some results which he had obtained, and which, as it seemed, could not be obtained easily by the method of series, and he asked for further See also:

• INFORMATION (from Lat. informare, to give shape or form to, to represent, describe)

information . Newton replied in a long letter to Oldenburg of the 24th of See also:

• OCTOBER

October 1676 . In this letter he gave a much See also:

• FULLER, ANDREW (1954-1815)
• FULLER, GEORGE (1822—1884)
• FULLER, M
• FULLER, MARGARET, MARCHIONESS
• FULLER, MELVILLE WESTON (1833-1910)
• FULLER, THOMAS (16o8-1661)
• FULLER, WILLIAM (1670--c. 1717)

fuller account of his binomial theorem and indicated a method of proof . Further he gave a number of results relating to quadratures; they were afterwards printed in the tract De quadratura curvarum . He gave many other results relating to the computation of natural logarithms and other calculations in which series could be used . He gave a general statement, similar to that in the letter to Collins, as to the kind of problems relating to tangents, maxima and minima, &c., which he could solve by his method, but he concealed his formulation of the calculus in an See also:

• ANAGRAM (Gr. ava, back, and ypa(beiv, to write)

anagram of transposed letters . The solution of the anagram was given eleven years later in the Principia in the words we have quoted from Wallis's Algebra .

In neither of the letters to Oldenburg does the characteristic notation of the fluxional calculus occur, and the words " fluxion " and " fluent occur only in anagrams of transposed letters . The letter of October 1676 was not despatched until May 1677, and Leibnitz answered it in See also:

• JUNE

June of that year . In October 1676 Leibnitz was in See also:

• LONDON

London, where he made the acquaintance of Collins and read the Analysis per aequationes, and it seems to have been supposed afterwards that he then read Newton's letter of October 1676, but he left London before Oldenburg received this letter . In his answer of June 1677 Leibnitz gave Newton a candid account of his differential calculus, nearly in the form in which he afterwards published it, and explained how he used it for quadratures and inverse problems of tangents . Newton never replied . 26 . In the Acta eruditorum of 1684 Leibnitz published a short memoir entitled Nova methodus See also:

• PRO

pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales Leibnitz's quantitates moratur, et singulare pro illis calculi genus . Differ- In this memoir the differential dx of a variable x, enu.al considered as the abscissa of a point of a curve, is said calculus. to be an arbitrary quantity, and the differential dy of a related variable y, considered as the ordinate of the point, is defined as a quantity which has to dx the ratio of the ordinate to the subtangent, and rules are given for operating with differentials . These are the rules for forming the differential of a See also:

• CONSTANT, BENJAMIN (1845-1902)

constant, a sum (or difference), a product, a quotient, a See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power (or See also:

• ROOT (late O.E. rot, adopted from Scand., cf. Norw. and Swed. rot, Dan. rod; the true O.E. word was wyrt, plant, represented in Ger. Wurz or Wurzel; the ultimate root is the same in both words, and is seen in Lat. radix)
• ROOT, ELIHU (1845– )

root) . They are equivalent to our rules (i.)-(iv.) of § 11 and the particular result d(x'") =mx"' -1dx . The rule for a function of a function is not stated explicitly but is illustrated by 'examples in which new variables are introduced, in much the same way as in Newton's Methodus fluxionum . In connexion with the problem of maxima and minima, it is noted that the differential of y is positive or negative according as y increases or decreases when x increases, and the discrimination of maxima from minima depends upon the sign of ddy, the differential of dy .

In connexion with the problem of tangents the differentials are said to be proportional to the momentary the project with regard to fluxions . In 168o Collins sought the assistance of the Royal Society for the publication of the tract, and this was granted in x682 . Yet it remained unpublished . The See also:

• REASON (Lat. ratio, through French raison)

reason is unknown; but it is known that about 1679, 1680, Newton took up again the studies in natural See also:

• PHILOSOPHY (Gr. gthos, fond of, and vo4 (a, wisdom)

philosophy which he had intermitted for several years, and that in 1684 he wrote the tract De motu which was in some sense a first draft of the Principia, and it may be conjectured that the fluxions were held over until the Principia should be finished . There is also reason to think that Newton had become dissatisfied with the arguments about infinitesimals on which his calculus was based . In the See also:

• PREFACE (Med. Lat. prefatia, for classical praefatio, praefari, to speak beforehand)

preface to the De quadratura curvarum (1704), in which he describes this tract as something which he once wrote (" olim scripsi ") he says that there is no See also:

• NECESSITY (Lat. necessitas)

necessity to intro-duce into the method of fluxions any See also:

• ARGUMENT

argument about infinitely small quantities; and in the Principia (1687) he adopted instead of the method of fluxions a new method, that of " See also:

• PRIME, PRIMER AND PRIMING

Prime and Ultimate Ratios." By the aid of this method it is possible, as Newton knew, and as was afterwards seen by others, to found the calculus of fluxions on an irreproachable method of limits . For the purpose of explaining his discoveries in See also:

• DYNAMICS
• DYNAMICS (from Gr. bbvayts, strength)

dynamics and See also:

• ASTRONOMY (from Gr. &arpov, a star, and v ,usiv, to classify or arrange)

astronomy Newton used the method of limits only, without the notation of fluxions, and he presented all his results and demonstrations in a geometrical form . There is no doubt that he arrived at most of his theorems in the first instance by using the method of fluxions . Further See also:

• EVIDENCE (Lat. evidentia, evideri, to appear clearly)

evidence of Newton's dissatisfaction with arguments about infinitely small quantities is furnished by his tract Methodus differentialis, published in 1711 by William Jones, in which he laid the See also:

• FOUNDATIONS

foundations of the " Calculus of Finite See also:

• DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

Differences." 24 . Leibnitz, unlike Newton, was practically a self-taught mathematician . He seems to have been first attracted to mathematics as a means of symbolical expression, and Leibnitz's on the occasion of his first visit to London, early in course of diseoYefy, 1693, he learnt about the doctrine of infinite series which See also:

• JAMES
• JAMES (Gr. 'IlrKw,l3or, the Heb. Ya`akob or Jacob)
• JAMES (JAMES FRANCIS EDWARD STUART) (1688-1766)
• JAMES, 2ND EARL OF DOUGLAS AND MAR(c. 1358–1388)
• JAMES, DAVID (1839-1893)
• JAMES, EPISTLE OF
• JAMES, GEORGE PAYNE RAINSFOP
• JAMES, HENRY (1843— )
• JAMES, JOHN ANGELL (1785-1859)
• JAMES, THOMAS (c. 1573–1629)
• JAMES, WILLIAM (1842–1910)
• JAMES, WILLIAM (d. 1827)

James Gregory, Nicolaus Mercator, Lord Brouncker and others, besides Newton, had used in their investigations . It appears that he did not on this occasion become acquainted with Collins, or see Newton's Analysis per aequationes, but he See also:

• PURCHASED

purchased Barrow's Lectiones .

On returning to See also:

• PARIS
• PARIS (also called ALEXANDROS)
• PARIS, ALEXIS PAULIN (1800-x881)
• PARIS, BRUNO PAULIN GASTON (1839-1903)
• PARIS, FRANCOIS DE (169o-1727)
• PARIS, LOUIS PHILIPPE ALBERT
• PARIS, TREATIES OF (1814-1815)

Paris he made the acquaintance of Huygens, who recommended him to read Descartes' Geometrie . He also read Pascal's Lettres de Deltonville, Gregory of St Vincent's Opus geometricum, Cavalieri's Indivisibles and the Synopsis geometrica of Honore Fabri, a book which is practically a commentary on Cavalieri; it would never have had any importance but for the See also:

• INFLUENCE (Late Lat. influentia, from influere, to flow in)

influence which it had on Leibnitz's thinking at this See also:

• CRITICAL (in alphabetical order of authors)

critical period . In August of this year (1673) he was at work upon the problem of tangents, and he appears to have made out the nature of the solution—the method involved in Barrow's differential triangle—for himself by the aid of a See also:

• DIAGRAM

diagram drawn by Pascal in a demon-station of the formula for the area of a spherical See also:

• SURFACE

surface . He saw that the problem of the relation between the differences of neighbouring ordinates and the ordinates themselves was the important problem, and then that the solution of this problem was to be effected by quadratures . Unlike Newton, who arrived at differentiation and tangents through integration and areas, Leibnitz proceeded from tangents to quadratures . When he turned his See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention to quadratures and indivisibles, and realized the nature of the process of finding areas by summing infinitesimal " rectangles, he proposed to replace the rectangles by triangles having a See also:

• COMMON

common vertex, and obtained by this method the result which we write In 1674 he sent an account of his method, called " transmutation," along with this result to Huygens, and early in 1675 he sent it to See also:

• HENRY
• HENRY (1129-1195)
• HENRY (c. 1108-1139)
• HENRY (c. 1174–1216)
• HENRY (Fr. Henri; Span. Enrique; Ger. Heinrich; Mid. H. Ger. Heinrich and Heimrich; O.H.G. Haimi- or Heimirih, i.e. " prince, or chief of the house," from O.H.G. heim, the Eng. home, and rih, Goth. reiks; compare Lat. rex " king "—" rich," therefore " mig
• HENRY, EDWARD LAMSON (1841– )
• HENRY, JAMES (1798-1876)
• HENRY, JOSEPH (1797-1878)
• HENRY, MATTHEW (1662-1714)
• HENRY, PATRICK (1736–1799)
• HENRY, PRINCE OF BATTENBERG (1858-1896)
• HENRY, ROBERT (1718-1790)
• HENRY, VICTOR (1850– )
• HENRY, WILLIAM (1795-1836)

Henry Oldenburg, secretary of the Royal Society, with inquiries as to Newton's discoveries in regard to quadratures . In October of 1675 he had begun to devise a symbolical notation for quadratures, starting from Cavalieri's indivisibles . At first he proposed to use the word omnia as an See also:

• ABBREVIATION (Lat. brevis, short)

abbreviation for Cavalieri's "sum of all the lines," thus writing amnia y for that increments of the abscissa and ordinate . A tangent is defined as a line joining two " infinitely " near points of a curve, and the " infinitely " small distances (e.g., the distance between the feet of the ordinates of such points) are said to be expressible by means of the differentials (e.g., dx) . The method is illustrated by a few examples, and one example is given of its application to " inverse problems of tangents." Barrow's inversion-theorem and its application to quadratures are not mentioned . No proofs are given, but it is stated that they can be obtained easily by any one versed in such matters . The new methods in regard to differentiation which were contained in this memoir were the use of the second differential for the discrimination of maxima and minima, and the introduction of new variables for the purpose of differentiating complicated expressions .

A greater novelty was the use of a letter (d), not as a symbol for a number or magnitude, but as a symbol of operation . None of these novelties account for the far-reaching effect which this memoir has had upon the development of mathematical analysis . This effect was a consequence of the simplicity and directness with which the rules of differentiation were stated . Whatever indistinctness might be felt to attach to the symbols, the processes for solving problems of tangents and of maxima and minima were reduced once for all to a definite routine . 27 . This memoir was followed in 1686 by a second, entitled De Geometria recondita et analysi indivisibilium atque infinitorum, Develop- in which Leibnitz described the method of using his ment new differential calculus for the problem of quadratures . of the This was the first publication of the notation fydx. calculus . The new method was called calculus summatorius . The See also:

• BROTHERS, RICHARD (1757-1824)

brothers See also:

• JACOB
• JACOB, JOHN (1812-1858)

Jacob (James) and Johann (John) See also:

• BERNOULLI

Bernoulli were able by 1690 to begin to make substantial contributions to the development of the new calculus, and Leibnitz adopted their word " integral " in 1695, they at the same time adopting his symbol " f." In 1696 the See also:

• MARQUIS

marquis de 1'See also:

• HOSPITAL (Lat. haspitalis, the adjective of hospes, host or guest)

Hospital published the first treatise on the differential calculus with the title Analyse See also:

• DES

des infiniment petits pour l'intelligence des lignes courbes . The few references to fluxions in Newton's Principia (1687) must have been quite unintelligible to the mathematicians of the time, and the publication of the fluxional notation and calculus by Wallis in 1693 was too See also:

• LATE

late to be effective . Fluxions had been supplanted before they were introduced . The differential calculus and the integral calculus were rapidly developed in the writings of Leibnitz and the Bernoullis .

Leibnitz (1695) was the first to differentiate a See also:

• LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)

logarithm and an exponential, and John Bernoulli was the first to recognize the property possessed by an exponential (az) of becoming infinitely great in comparison with any power (20°) when xis increased indefinitely . See also:

• ROGER (d. 1139)
• ROGER (d. 1181)

Roger See also:

• COTES, ROGER (1682-1716)

Cotes (1722) was the first to differentiate a trigonometrical function . A great development of infinitesimal methods took place through the See also:

• FOUNDING (from Lat. fundere, to pour)

founding in 1696–1697 of the " Calculus of See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

Variations " by the brothers Bernoulli . 28 . The famous dispute as to the priority of Newton and Leibnitz in the invention of the calculus began in 1699 through Dispute the publication by See also:

• NICOLAS
• NICOLAS, SIR NICHOLAS HARRIS (1799–1848)

Nicolas Fatio de Duillier of a See also:

• CON

con- tract in which he stated that Newton was not only the cerning first, but by many years the first inventor, and insinu-Priority. ated that Leibnitz had stolen it . Leibnitz in his reply (Acta Eruditorum, 1700) cited Newton's letters and the testimony which Newton had rendered to him in the Principia as proofs of his See also:

• INDEPENDENT

independent authorship of the method . Leibnitz was especially hurt at what he understood to be an endorsement of Duillier's attack by the Royal Society, but it was explained to him that the apparent approval was an See also:

• ACCIDENT
• ACCIDENT (from Lat. accidere, to happen)

accident . The dispute was ended for a time . On the publication of Newton's tract De quadratura curvarum, an See also:

• ANONYMOUS

anonymous See also:

• REVIEW (Fr. revue, from revoir, to see again, Lat. re and videre)

review of it, written, as has since been proved, by Leibnitz, appeared in the Acta Eruditorum, 1705 . The anonymous reviewer said: " Instead of the Leibnitzian differences Newton uses and always has used fluxions . . . just as Honore Fabri in his Synopsis Geometrica substituted steps of movements for the method of Cavalieri." This passage, when it became known in See also:

• ENGLAND
• ENGLAND, THE CHURCH OF

England, was understood not merely as belittling Newton by comparing him with the obscure Fabri, but also as implying that he had stolen his calculus of fluxions from Leibnitz . Great indignation was aroused; and John Keill took occasion, in a memoir on central forces which was printed in the Philosophical Transactions for 1708, to affirm that Newton was without doubt the first inventor of the calculus, and that Leibnitz had merely changed the name and mode of notation .

The memoir was published in 1710 . Leibnitz wrote in 1711 to the secretary of the Royal Society (Hans See also:

• SLOANE, SIR HANS (1660-1753)

Sloane) requiring Keill to retract his See also:

• ACCUSATION (Lat. accusatio, accusare, to challenge to a causa, a suit or trial at law)

accusation . Leibnitz's letter was read at a See also:

• MEETING (from " to meet," to come together, assemble, 0. Eng. metals ; cf. Du. moeten, Swed. mota, Goth. gamotjan, &c., derivatives of the Teut. word for a meeting, seen in O. Eng. Wit, moot, an assembly of the people; cf. witanagemot)

meeting of the Royal Society, of which Newton was then See also:

• PRESIDENT (Fr. president, from Lat. praesidens, post-Augustan Lat. for praeses, director, ruler, from praesidere, to sit in front of, preside)

president, and Newton made to the society a statement of the course of his invention of the fluxional calculus with the dates of particular discoveries . Keill was requested by the society " to draw up an account of the See also:

• MATTER

matter under dispute and set it in a just light." In his See also:

• REPORT (O.Fr. report or raport, modern rapport, from O.Fr. reporter, mod. rap porter, Lat. reportare, to bring back, in poetical use only, of bringing back an account, news, &c.)

report Keill referred to Newton's letters of 1676, and said that Newton had there given so many indications of his method that it could have been understood by a See also:

• PERSON, OFFENCES AGAINST THE

person of ordinary intelligence . Leibnitz wrote to Sloane asking the society to stop these unjust attacks of Keil], asserting that in the review in the Acta Eruditorum no one had been injured but each had received his due, submitting the matter to the See also:

• EQUITY (Lat. aequitas)

equity of the Royal Society, and stating that he was persuaded that Newton himself would do him See also:

• JUSTICE (Lat. justitia)

justice . A See also:

• COMMITTEE (from commute, an Anglo-Fr. past participle of commettre, Lat. committere, to entrust; the modern Fr. equivalent comite is derived from the Eng.)

committee was appointed by the society to examine the documents and furnish a report . Their report, presented in See also:

• APRIL

April 1712, concluded as follows: " The differential method is one and the same with the method of fluxions, excepting the name and mode of notation; Mr Leibnitz calling those quantities differences which Mr Newton calls moments or fluxion, and marking them with the letter d, a See also:

• MARK
• MARK, CARL (1858– )
• MARK, GOSPEL OF ST
• MARK, ST

mark not used by Mr Newton . And therefore we take the proper question to be, not who invented this or that method, but who was the first inventor of the method; and we believe that those who have reputed Mr Leibnitz the first inventor, knew little or nothing of his See also:

• CORRESPONDENCE
• CORRESPONDENCE (from med. scholastic Lat. correspondentia, corrcspondere, compounded of Lat. cum, with, and respond ere, to answer; cf. Fr. correspondance)

correspondence with Mr Collins and Mr Oldenburg long before; nor of Mr Newton's having that method above fifteen years before Mr . Leibnitz began to publish it in the Ada Eruditorum of See also:

• LEIPZIG

Leipzig . For which reasons we reckon Mr Newton the first inventor, and are of See also:

• OPINION (Lat. opinio, from opinari, to think)

opinion that Mr Keill, in asserting the same, has been no ways injurious to Mr Leibnitz." The report with the letters and other documents was printed (1712) under the title Commercium Epistolicum D . Johannis Collins et aliorum de analysi promota, jussu Societatis Regiae in lucem editum, not at first for publication . An account of the contents of the Commercium Epistolicum was printed in the Philosophical Transactions for 1715 .

A second edition of the Commercium Epistolicum was published in 1722 . The dispute was continued for many years after the death of Leibnitz in 1716 . To translate the words of See also:

• MORITZ, KARL PHILIPP (1757-1793)

Moritz Cantor, it " redounded to the discredit of all concerned." 29 . One lamentable consequence of the dispute was a severance of See also: