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■feems to be the chief caufe* why to fuppofe the infinite divifibility of finite Extenfion hath been thought neceflaryin geometry. The feveral abfurdkies and contradictions which flowed from this falfe principle might* one would think* have been efteemed fo many demonftrations againftit; but, by I know not what logic, it is held that proofs a pofteriori are not to be admitted againft proportions relating to infinity. As though it were nut impoflible even for an infinite mind to reconcile contra- dictions j or, as if anything abfurd and repugnant could have a neceffary connection with truth, or flow from it. But who- ever confiders the weaknefs of this pretence, will think it was contrived on purpofe to humour the lazinefs of the mind, which had rather acquiefce in an indolent fcepticifm, than be at the pains to go thro' with a fevere examination of thofe principles it hath ever embraced for true.
Of late the fpeculations about infinities have run fo high, and grown to fueh Arrange notions, as have oceafioned no fmall fcruples and difpute among the geometers of the prefent age : Some are of great note, who not content with holding that finite lines may be divided into an infinite number of parts, do yet farther maintain, that each of thofe infinitefimals is itfelf fubdivifible into an infinity of other parts, or infinitefi- mals of a fecond order, and fo on ad infinitum. Thefe, I fay, aitert there are infiniteftmals of infinitefimals of infinitefimals, without ever coming to an end. So that according to them an inch doth not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts. Others there be who hold all orders of infinitefimals below the firft to be nothing at all, thinking it with good rea- son abfurd, to imagine there is any pofitive quantity or part of Extenfion, which, though multiplied infinitely, can never equal the fmalleft given Extenfion. And yet, on the other hand, it feems no lefs abfurd, to think the fquare, cube, or other power of a pofitive real root, mould itfelf be nothing at all ; which they who hold infinitefimals of the firft order, denying all of the fubfequent orders, are obliged to maintain. Have we not therefore reafon to conclude, that they are both ■ in the wrong, and that there is in effect no fuch thing as parts infinitely fmall, or an infinite number of parts contained in any finite quantity ? But you will fay, that if this doctrine obtains, it will follow the very foundations of geometry are deitroyed : And thofe great men who have raifed that faience to fo aitonifhing an height, have been all the while building a caftle in the air. To this may be replied, that whatever is ufeful in geometry and promotes the benefit of humane life, doth ftill remain firm and unlhaken on our principles. That fcience confidered as practical, will rather receive advantage than any prejudice from what hath been faid : But to fet this in a due light, may be the fubject of a diftincf. inquiry. For the reft, tho* it fhould follow that fome of the more intricate and fubtil parts of fpeculative mathematics may be pared off without any prejudice to truth ; yet I do not fee what damage will be thence derived to mankind. On the contrary, it were highly to be wifhed, that men of great abilities and obftinate application would draw off their thoughts from thofe amufe- ments, and employ them in the ftudy of fuch things as lit nearer the concerns of life, or have a more direct influence on the manners.
If it be faid, that feveral theorems undoubtedly true are dif* covered by methods in which infinitefimals are made ufe of, which could never have been, if their exiftence included a contradiction in it. I anfwer, that upon a thorough exami- nation it will not be found, that in any inftance it is neceflary to make ufe of or conceive infmitefimal parts of finite lines, or even quantities lefs than the minimum fenfibile : Nay, it will be evident this is never done, it being, impoflible. Thus far D. Berkeley.
On the other hand it is obferved by an eminent mathemati- cian, that geometricians are under no neceffity of fuppof- ing that a finite quantity or Extenfion confifts of parts infinite in number, or that there arc any more parts in a given mag- nitude than they can conceive or exprefs : It is fufficient that it may be conceived to be divided into a number of parts equal to any given or propofed number, and this is all that is fup- Jwfed in ftrict geometry concerning the divifibility of magni- tude. It istiuc, that the number of parts, iuto which a given magnitude may be conceived to be divided, is not to be fixed or limited, becaufe no given number is fp great but a greater than it may be conceived and affigned : But there is not there- fore any neceffity for fuppofing that number infinite ; and if fome may have drawn very abftrufe confequences from fuch fuppofitions, geometry is not to be loaded with them. See Nli.Madaurin's Treatife of Fluxions, Art. 290. Though geometricians are under no neceffity of fuppofing a given magnitude to be divided into an infinite number of parts, or to be made up of infinitefimals, they cannot fo well avoid fuppofing- it divided into a greater number of parts than may be difHnguifhed in it by fenfe in any particular determi- nate circumftances. But they find no difficulty in conceiving this ; and fuch a fuppofition does not appear repugnant to the common fenfe of mankind, but on the contrary to be moft agreeable to it, and to be illuft rated by common obfervation. It would fcem very unaccountable, not to allow them to con- ceive a given line, of an inch in length for example, viewed Suppi.. Vol, I.
at -the diftance of ten feet, to be divided into more parts that! are difcerned in it at that diftance ; fince by bringing it nearer a greater number of parts is actually perceived in it; 'Not is it eafy to limit the number of parts that be perceived in it when it is Drought near to the eye, and is feen through a little hole in a thin plate; or when by any other Contrivance it is ren- dered diftinct at fmall diftances from the eye. If we conceive a given line that is the object of fight to be divided into more parts than we perceive in it, it would feem that no good rea- fon can be affigned, why we may not conceive tangible mag- nitude to be divided into more parts than ate perceived in it by the touch* or a line of any kind to be divided into any given number of parts, whether fo many parts be actually dif- tinguifhed by fenfe* or not. If the hyperbola and its afymp- tote were accurately defcribed, they would feem to fenfe to join each other, at various diftances from the center, accord*- ing to the different circumftances in which they might be per- ceived ; but we may conceive the ordinate at the point where they feem to join to have a real magnitude, in the fame man- ner as we conceive a given line to fubfift when it is carried to fo great a diftance that it vanifhes to fight, or any fmall par- ticle (as an atom in the fun-beams) to cxift, thV it cfcape the touch, or have no tangible magnitude. It may perhaps illuftrate this, if it be it confidered* that the curve cannot be faid to meet its afymptote in this cafe, in the fame fenfe that a circle is faid to meet its diameter* which it appears to inter- feet in all cafes, whatever the diftance or pofition of the fi- gure, or the acutenefs of the fenfe may be ; whereas the or- dinate of the hyperbola that vanifhes to fight at a great dif- tance, becomes vifible at a lefs diftance ; and may be diftin- guifhed into more and more vifible parts, in proportion as it approaches to the eye, or the fenfe is more acute; And furely it muff, be allowed that there is ground for a difference between a line that efcapes the fight and vanifhes, becaufe of its diftance from the eye, and a line that in no cafe can ever be perceived, or can be fuppofed to have any exiftence. Per- haps it will be faid by fome, that, ftrictly fpeaking* it is not the fame line that in thofe different circumftances has a greater and lefs number of vifible parts. In anfwer to this, it Is fufficient for ourpurpofeto obferve, that as there can hardly be any philofopher but will allow that there is fome fenfe in which it is the fame inch- line that has morevifible parts at eight inches diftance from the eye than when it is held at the length of the arm; fo it is not incumbent on us to explain in what fenfe this is to be underftood according to every fcheme of philofophy: It is enough that this fenfe rnuft be fuppofed to be plain and obvious, as it is univerfal, and that geometricians ought to be allowed to con- fider lines and figures in this fenfe as well as every body eKe* Philofophers and the vulgar equally conceive the fun and pla- nets, and the other objects of their obfervation and enquiries, to be the fame bodies, when feen at different diftances or dif- ferent times : And if they were not allowed to confider thofe bodies as made up of more parts than are perceived by fenfe, and geometricians were under the fame limitations as to mag- nitude in general, they would not be a little perplexed ; nor is it the more intricate and fubtile part of thofe fciences only that would be thus pared off. The learned author above-men- tioned tells us, " That the magnitude of the object which " exifts without the mind, and is at a diftance, continues al- " ways invariably the fame 3 ." He feems to fpeak of tangi- ble magnitude. It is not our bufinefs here to enquire how, according to his doctrine, tangible magnitude can be con- ceived to exift without the mind any more than vifible mag- nitude. This conceffion perhaps is made only for tiie fake of his argument in this place ; but the evidence for the exiftence of fuch an object may very well be fuppofed to approach to that which we have for the exiftence of any other objects that are not immediately perceived by us. And fince he admits it, and argues from it, in this treatife ; it would feem that fome invariable magnitude is to be allowed, which we apprehend by the fight, though not immediately ; and that this magni- tude may be conceived to be divided into any given number of parts, from the demonftrations propofed by geometricians on this fubject b . In applying which, it ought to be remem- bered, that a furface is not confidered by them as a body of the leaft fenfible magnitude, but as the termination or boundary of a body ; a line is not confidered as a furface of the leatfc fenfible breadth, but as the termination or limit of a furface : Nor is a point confidered as the leaft fenfible line, or a mo- ment as the leaft perceptible time ; but a point as a termina- tion of a line, and a moment as a termination or limit of time. In this fenfe they conceive clearly what a furface, line, point, and a moment of time is c ; and the pojiulata of Eu- clid being allowed and applied in this fenfe, the proofs by which it is fhewn that a given magnitude may be conceived to be divided into any given number of parts, appear Satisfactory : And if we avoid the fuppofing the parts of a given magnitude to be infinitely fmall, or to be infinite in number, this feems to be all that the moft fcrupulous can require. d — [ a New Theory of Vifion, §. 55. b Ibid. §. 54. c The Analyft, §. 31. d Mr. Madaurin's, Treat, of Fluxions, Art. 290, 291.] Extension offraclured Limbs, in furgery. When the fractured bones maintain their natural fituation, the furgeon has nothing to do but to apply a proper bandage to keep them in it j but when 10 O the
