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following axioms are as evident, as that a greater or lefs fpice is defcribed in a given time, according as the velocity of the motion is greater or lefs.
Axiom I. The fpace defcribed by an accelerated motion is greater than the fpace which would have been defcribed in the fame time, if the motion h:id not been accelerated, but had continued uniform from the beginning of the time. Axiom 2. The fpace defcribed by a motion while it is accele- rated, is lefs than the fpace which is defcribed in an equal time by the motion that is acquired by that acceleration con- tinued uniformly.
Axiom 3. The fpace defcribed by a retarded motion is lefs than the fpace which would have been defcribed in the fame time, if the motion had not been retarded, but had continued uniform from the beginning of the time.
Axiom 4. The fpace defcribed by a motion while it is retard- ed, is greater than the fpace which is defcribed in an equal time by the motion that remains after that retardation, con- tinued uniformly.
From thefe axioms general theorems concerning motion, of ufe in the doctrine of Fluxions, may be demon ft rated. Thus when the fpaces defcribed by two variable motions are always equal, or in a given ratio, the velocities are always equal, or in the fame given ratio ; and converfely, when the velocities of two motions are always equal to each other, or in a given ra- tio, the fpaces defcribed by thofe motions in the fame time are always equal, or in that given ratio : that when a fpace is al- ways equal to the fum or difference of the fpaces defcribed by two other motions, the velocity of the firft motion is always equal to the fum or difference of the velocities of the other motions ; and converfely, that when a velocity is always equal to the fum or difference of two other velocities, the fpace defcribed by the firft motion is always equal to the fum or difference of the fpaces defcribed by thefe two other motions. See Mr. Maclaurin's Treatife of Fluxions, Book I. chap. 1, The main point in the method of Fluxions, is to obtain the Fluxion of the rectangle, or product of two indeterminate quan tities, fince from thence may be derived rules of all other pro- duces and powers be the coefficients, or the indices what they will, integers or fractions, rational or furd; according to the manner of Sir Ifoac Newton in the fecond lemma of his fe cond book of principles.
Mr. Maclaurin has therefore been very full in eftablifhing this point ; and after what he has faid, we prefume that no reafo- nable objection can lie either againft the clearnefs and dif ■ tinctnefs of the notion of Fluxions, or againft the truth of the principles, or accuracy of the demonft rations by which their meafures are determined. We cannot here infert his demon- ftrations at length ; but as many readers may, perhaps, be defirousof feeing the argument contracted into a narrow com- pafs, we (hall here give a fummary of it, from the Philof. Tranf. N°. 468. p. 331.
A triangle that has two of its fides given in pofition, is fup- pofed to be generated by an ordinate moving parallel to itfelf along the bafe. When the bafe increafes uniformly, the tri- angle increafes with an accelerated motion, becaufe its fuc- ceffive increments are trapezia, that continually increale ; therefore if the motion with which the triangle flows, was continued uniformly from any term for a given time, a lefs fpace would be defcribed by it, than the increment of the tri- angle, which is actually generated in that time by axiom 1 ; but a greater fpace than the increment which was actually ge- nerated in an equal time preceeding that term, by axiom 2. And hence it is demonftratcd, that the Fluxion of the triangle is accurately meafured by the rectangle contained by the cor- refponding ordinate of the triangle, and the right line which meafures the Fluxion of the bafe. The increment which the triangle acquires in any time, is refolved into two parts ; that which is generated in confequence of the motion with which the triangle flows at the beginning of the time, and that which is generated in confequence of the acceleration of this motion for the fame time. The latter is juftly neglected in meafuring that motion (or the Fluxion of the triangle at that term) but may ferve for meafuring its acceleration, or the fe- cond Fluxion of the triangle. The motion with which the triangle flows, is fimilar to that of a body defcending in free fpaces by an uniform gravity, the velocity of which, at any term of the time, is not to be meafured by the fpace defcribed by the body in a given time, either before or after that term, becaufe the motion continually increafes, but by a mean be- tween thefe fpaces.
When the fides of a rectangle increafe or decreafe with uni- form motions, it may be always confidered as the fum or dif- ference of a triangle, and trapezium ; and its Fluxion is derived from the laft propofition. If the fides increafe with uniform motions, the rectangle increafes with an accelerated motion ; and in meafuring this motion at any term of the time, a part of the increment of the rectangle that may be determined a, is rejected, as generated in confequence of the acceleration of that motion, [a See Marfaur. I. c. art. J02.J Thofe who have well underftood what precedes, will not be a; a lofs to conceive, that the Fluxions of a curvilineal area, S u p p l. Vol. It
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whether generated by an ordinate moving parallel to itfelf, bf by a radius revolving round a given center, may be determin- ed by demonftrations of the lame kind. When the ordinate's of the hgure increafe, the increment of the area may be re- folved m like manner into two parts, one of which only is to be retained in meafuring the Fluxion of the area, the other be- ing rejecT:ed as generated in confequence of the acceleration 6f the motion with which the figure flows. What has been hitherto faid will let the difference between the notions of Fluxions and that of infinitefimals, in a clear ?, \ Fhmms ma y always be rc'prefentcd by finite quantities, i he fuppofition of an infinitely little magnitude is too bold a poltulatum for fuel) a fcience as geometry. Nor have authdrs accounted explicitly for the truth and perfed accuracy of the conclufions derived from this conlideration. When they de- termine what is called the difference, but more accurately the Fluxion of a quantity, they tell us, they reje£t certain parts of the element, becaufe they become infinitely lefs than the other paits. But this is no proper reafon, not only becaufe a proof of this nature may leave fome doubt as to the accuracy of the conclufion; but becaufe it may be demonftrated that thefe parts ought to be neglefled by them at any rate, or that it would be an error to retain them. If an accountant, that pretends to a fcrupulous exaanefs, fhould tell us, he had ncg- lefled certain articles, betaufe he found them to be of final! importance ; and it mould appear that they ought not to have been taken into confideration by him on that occafion, but be- long to a different account, we mould approve his conclufions as accurate, but not his reafons. See Maclauriu's Treatife of Fluxions in the Preface, and book I. ch. 12. where the methud of infinitefimals is exprefsly treated of. See alio the article Infinitesimal
Mr. Maclaurin in the firft part of his treatife confiders Fluxions in a merely geometrical form, and has demonftratcd the rules of the method with all poffible accuracy and rigour; but as the great improvements made by this doflrine are chiefly to be afcribed to the facility, concifenefs, and great extent of 'the methods of computation, or the algebraic part, it is necefiary to add fome account of thefe methods alfo. Any quantities produced from each other by an algebraic ope- ration, or whole relatibn is expreffed by any algebraic form, being fuppofed to increafe or decreafe together, fome will be found to increafe or decreafe by greater differences, or at a great- er rate, others by lets differences or at a lefs rate ; and while fome are fuppofed to increafe or decreafe at one conltanf rate by equal fucceffive differences, others increafe or decreafe by differences that are always varying. Thefe rates of increafe or decreafe may be determined by comparing the velocities of points that always defcribe lines proportional to the quantities as before mentioned, but they may alfo be determined without having recourfe to fuch fuppolitions, by a juft reafoning from the limultaneous increments or decrements themfelves. When a quantity A increafes by differences equal to a,2Aincrea- fes by differences equal to 2a, and manifeftly increafes or decreafes at a greater rate than A in the proportion of 2 a to a or 2 to i ;
and if m and n be invariable '" A .
increafes or decreafes by dif-
ferences equal to "It and therefore at a greater or lefs rate n
than A in proportion as 11 is greater or lefs than a, or m
n is greater or lefs than n. This feems to be eafily conceived, without having recourfe to any other confiderations, than the relation of the differences by which the quantities increafe or de- creafe. In order therefore to avoid figurative expreflions in the algebraic part, it will be proper to fubftitute in the place of the definition and axioms before mentioned, others that are rather of a more general import, but are perfeftly confident with them. Thus,
Fluxions of quantities are any meafures of their rejpcclive rates of increafe, or decreafe, while they vary or flow together. There can be no difficulty in determining thofe meafures when the quantities increafe ot decreafe by fucceffive differences that are always in the fame invariable proportion to each other, while A by increafing becomes equal to A -t- a, or by decreaf- ing equal to A — a, 2 A becomes equal to 2 A -f- 2 a, or to 2 A — ■ 2 « ; and as 2 A increafes or decreafes at a greater rate than A in the proportion of 2 a to a ; fo the Fluxion of A be- ing fuppofed equal to a, the Fluxion of 2A muft be equal to
2 a. In the fame manner the Fluxion of -^ x A (or of — x
n n
A -j- e fuppofing m, n and e to be invariable) is — X a ; and fince
m may be to n in any affignable ratio, a quantity may be al- ways affigned that {hall increafe or decreafe, at a greater or lefs rate than A in any proportion, or that {hall have its Fluxion greater or lefs than the Fluxion of A in any ratio. In fuch cafes the ratio of the Fluxions and that of the differences by which the quantities increafe or decreafe are the fame. But while A is fuppofed to increafe at a conftant rate by any equal fucceffive differences, if B increafe or decreafe by diffe- n N rences
