It is no hard matter to find a Method of railing in us iuch Paflions as we defiie, fince the Knowledge we have given in the foregoing Books of the Union betwixt Soul and Body has fuf- fjciently open'd the way to it: In a word, no more is requir'd, than to think attentively upon thofe Obje&s, that by the Inftitution of Nature are able to raife the Paffions. Thus we may al- moft at any time excite in our Hearts whatever Paffion we have occafion for $ but becaufe we can eafier excite them at any time than fupprefs them, or remedy the Diforders they caufe in the Ima- gination, we muft be very lober and cautious in employing them.
Above all, we muft take care not to judge of Things by Paffion, but only by the clear Sight of the Truth, which is almoft impoffible when the Paflions are fomewhat lively 5 they ought only to raife our Attention, but they nevet fail of ftirring up their proper Ideas, and violently driving the Will to judge of Things by thofe Ideas that affecf it, rather than by the pure and abftrafted Ideas of Truth, that make no Impreffion upon it: So that we often make Judgments which laft no lon- ger than the Paffion, becaule they are not produced by the clear Sight of the immutable Truth, but by the Circulation of the Blood.
True it is that Men are wonderfully obftinate in fome Errours, which they maintain as long as they live; but then thofe Errours have other Caufes than the Paffions, or at leaft depend on fuch as are permanent and Lifting, proceeding from the Conftitution of the Body, from Intereft, or from ibme other durable Caufe. For Inftance -, Intereft being a Motive of a continual ftanding, produ- ces a Paffion that never dies; and the Judgments that arife from it are very long liv'd. But all the other Sentiments of Men, which depend upon particular Paffions, are as inconftant as the Fermen* ration of their Humours: They fay one while this, another while that •, and yet what they fay is commonly conformable to what they think: And as they run from one counterfeit Good to another, by the Motion of their Paffion, and are difgufted at it when that Motion ceafes; fo they run from one falfe Syfiem into another, and ardently affert a falfe Opinion, when Paffion makes it probable y which, the Paffion ebbing, they afterwards forfake. By their Paffions they tafte of every Good, without finding any really fo -, and by the fame Paffions fee all Truths, without difcovering any thing abfolutely true •, though in the time of their Paffion, what they tafte feems to them the Sovereign God, and what they fee an undeniable Truth.
The Senfes are the fecond Spring, whence we can draw Succours to make the Mini attentive. Sen/at ions are the very Modifications of the Soul, and differ from the pure Ideas of the Mind; the former railing a much ftronger Attention than the latter. So that 'tis plain, that to fupply the want of Application to infenfible Truths, it may be fit to exprefs them in a fenfible and moving manner.
'Tis for that Reafon, that Geometricians exprefs by fenfible Lines, the Proportions that are be- twixt f everal Magnitudes -, for by drawing Lines upon Paper, they draw, as I may fay, anfwerable Ideas upon their Mind, and make them more familiar by Seeing them at the fame time that they Conceive them. Thus feveral very difficult Things may be taught to Children, though they be not fufceptible of abftrafted Truths, by reafon of the Nicety of the Fibres of their Brain: Their Eyes fee nothing but Colours, Pictures, Images; but their Mind confiders the Ideas that anfwer thofe fenfible Objects.
But we muft take a fpecial Care not to over-fhadow the Objeffs which we will confider or repre- fent to others, With fo much Senfibility, that the Mind mould be more taken up with it, than with the Truth it felf* which is a moft confiderable and common Fault $ for we meet every day with Men that apply themfelves only to what moves the Senfes, and exprefs themfelves in fuch a fen- fible manner, that Truth is as itifled under a vain and pompous Apparel of their falfe Eloquence $, fo that their Hearers, being more affecfed with the Meafure of their Periods, and the Motions of their Figures, than by the Reafons they alledge, give way to be perfuaded, without fo much as knowing what caufes their Perfuafion, or what they are perfuaded of.
And therefore we muft fo carefully moderate the Senfibility of our Expreffions, as only juft to make the Mind attentive. There is nothing more beautiful than Truth; neither can we pretend to make it handfomer, by daubing it with fenfible Colours, that have no Solidity in them, and are pleafing but a fhort time. We might perhaps make it more fine and delicate, but fhould enerve and emafculate it: So that we ought not to fet it off with fo much Luftre and Brightnefs, that the Mind be more taken up with the Ornaments, than with the Body it felf-, this being to deal with it as fome Perlbns do with themfelves, when loaded with fuch abundance of Gold and pre- cious Stones, they appear the leaft confiderable part of the whole which they make up with their Clothes. We muft drefs the Truth as are thofe Magiftrates of Venice, who are oblig'd to wear a plain Gown and a Cap, to diftinguifh them from the Commonalty-, that Men may look on their Faces With Reverence and Attention, without admiring their Apparel. Laftly, We muft take care not to furcharge it with too great a Retinue of delightful Things, that diffipate the Mind, and ob- ftruft its View, left we fhould give to any thing elfe the Honours due to it: As it often happens to Princes, who cannot be diftinguHh'd amonglt the great Number of their Courtiers and Atten- dants, who affume to themfelves that Air of Greamefs, and Majeftica.1 Countenance, which only becomes the Sovereign? themfelves. But Chap. IV. The Search after Truth.
But to give a more confiderabk: Inftance, I fay, that Truth muft be propofed to others, as it manirefts it felf. The Sight of Men, fihce the Fait of their Fore-fathers, is too weak to look on Truth it felf, and therefore Sovereign Truth has made it felf fenfible by coming, inverted with our Humanity, that it might attraO: our Thoughts, enlighten our Mind, and appear lovely to our Eyes. So we may, according to that Pattern, adorn with fomething ienfible the Truths we endeavour to underftand our felves, and to teach others, that we may fix the Mind upon them, which loves what is ienfible, and is not eafily delighted by Things that flatter not the Senfes. The Eternal Wifdom has made it felf ienfible, but not glittering and pompous \ becoming fenfible, not to faften us to what is fenfible, but to raife us to what is intellectual, and to condemn and facrifice Senfibility in his own Perfon. So we muft make ufe, in the Knowledge of Truth, _ of fomething fenfible, but not too fplendid h that cannot indear too much the fenfible Obje£t, but only. keep.open the Eye of our Mind in the Contemplation of mere intellecFual Truths:. Such Senfibility mould be employ 'd, as we may dimpate, annihilate, and willingly facrifice upon the Sight of the Truth, to which it has conduced us. The Eternal Wifdom has offer'd it felf to us from without, in a fenfible man- ner, not to keep us abroad, but that we may retire within our felves, and that the Inner Mart might intellectually confider it: So we muff, in our Search of Truth, make ufe of fomething fen- fible, which may not keep us abroad gazing on its Luftre, but make u$ enter into our felves, and Ihengthen our Attention and Union to the Eternal Truth, Which only is able to rule the Mind, and enlighten it upon any Subject whatfoever.
4* CHAP. IV.
Of the Vfe of Imagination to wahg the Mind attentive, and efpecially of the ZJfefulnefs of Geometry.
WE had need be very circumfpecVand cautious in the Choice and Ufe of thofe Helps that we may draw from our Senfes and Paflions, to become attentive to the Truth 5, becaufe our Sen- fes and Paihons too vividly affe£t us, and fo much fill up the Capacity of the Mind, that it often lees nothing but its own Senfations, when it propofes to diicover Things in their own Nature. But as to thofe Succours which our Imagination may afford us, they make the Mind attentive, without fruitlefly dividing its Capacity, and wonderfully help us to a clear and diftinft Perception of Ob- jefts; fo that they are for the moft part very ufeful, as will be made plain by fome Inftances.
We know that a Body is moved by two or ieveral different Caufes, towards two or feveral dif- ferent Places, whereunto it is equally or unequally driven by thefe Forces j that the Force of the Motion perpetually increafes or decreafes, according to fome known Proportion. We are asked what way that Body goes, in what place it fhall be at fuch or fuch a Moment, with what degree of Celerity it fhall be> endued when 'tis come to fuch a place j and other like Queftions* i. From the point A, whence we fuppofe that it be- gins to move, draw the indefinite Lines AB, AC, that make the Angle BAC, if they cut each other •, for AB and AC are direct, and cut not each other when the Motions they exprefs are dire£lly oppofite. Thus we diftincTtly represent to the Imagination, or, if you pleafe, to the Senfes, the way that Body mould take when it is only moved by one of thefe Forces, either towards B, 2. But if the Force that moves it towards B be equal to that which moves it towards C, then divide the Lines AB and AC into the parts i, 2, 3, 4. 1, 11, in, iv, equally diftant from A: If the Force that moves it to- wards B be double of that which moves it towards C, take in the Line AB Parts that are double of thofe that you cut in AC: If that Force be fubduple, take them fubduple j if it be thrice greater or leffer, cut them like- wife thrice greater or leffer •, and fo proportionably. The Divifions of thofe Lines will represent to the Ima- gination the different Degrees of thofe moving Forces, and withal, the Space that they fhall caufe the Body to run over.
3. Draw through thofe Divifions Parallels upon AB and AC, to have the Lines 1 X, 2X, 3 X, &c. equal to A 1, A 1 1, A 1 1 1, &c. and 1 X, 1 1 X, 1 1 1 X, equal to A 1, A 2, A 3, £?c. that reprefent the Spaces through which thofe Forces carry that Body. Through the In- terferons of thofe Parallels draw the Line AXYE, that reprefent to the Imagination, firft, the trueGreatnefs of the compofed Motion of that Body which is fuppofed B v y ¥~ I 1 5 } / E c K M ALEBRANCHE ConC&Th ung Book VI.
iv nr it i A A c to be driven at the fame time towards B and C, by two different Forces, according to fome certain Proportion: Secondly ', The Way that it is to pais through: And, /aftly, All the 'Places in which it mult be in a determinate Time. So that this Line lerves not only to bear up the Sight of the Mind in the In- quiry after all the Truths that are difcoverable in the Queftion propofed; but alfo repreients the Solution of it in a fenfible and convincing manner.
Firft, That Line A X Y E expreffes the true De- grees* of the compound Motion: For we fenfibly perceive, that if each of the Forces which produce it can promote the Body a Foot in a Minute, its compofed Motion will be of two Foot in a Minute, if both moving Forces do perfectly agree -, flnce in that Cafe it is enough to add A B to AC. But if thofe Forces are not altogether equal, the compofed Motion A E will be greater than one of the Com- poundings, AB or AC, by the Line YE: Where-*' as if thofe Motions be oppofite in any thing, the compofed will be lefler than either of the com-' pounding, by the Line YE; and if they be entirely oppofite, it will come to nothing.
Secondly, The Line A X Y E feprefents to the Imagination the Way which that Body mail go': For we fenfibly perceive in what Proportion it mall advance more to one than to the other fide. We likewife perceive, that all the compound Motions are direct, when each of the compounding is always the fame, though they be unequal betwixt themfelves •, or when the Compoundings are always equal betwixt themfelves, though they be not conftantly the fame. Laftly, It plainly appears, that the Lines defcribed by thofe Motions are crooked, when the Compounding are both unequal to each other, and not always the fame.
Thirdly, Laft of all, That Line reprefents to the Imagination all the Places in which that Body, driven by two different Forces towards two different Places, mail be found 5 fo that we can pre- cifely mark the Point in which that Body (hall be in any Inftant whatfoever. For inftance, If you defire to know in what Place that Body mall be at the beginning of the fourth Minute, divide the Lines AB, or AC, in fuch Parts as exprefs the Space through which thofe known Forces might each of them carry that Body within a Minute •, take three of thole Parts in either of thefe Lines, then draw through the beginning of the fourth 3 X parallel to AB, or 111 X pa- rallel to A C^ for 'tis evident that the Point X, which either of thole Parallels determine in the Line AX YE, defigns the Place in which that Body fhall be at the beginning of the fourth Mi- nute of its Motion* Thus that Method of examining Queftions, not only keeps up the View of the Mind, but alfo affords the Solution of them, and withal a fufficient Light to difcover unknown Things by a few that are known.
For Inftance: After what has been faid, it is e-" nough only to know, that a Body that was in A at fuch a time, is in E at fuch another •, and that the different Forces that drive it, defcribe Lines that make fuch an Angle as B A C, to difcover the Line of its compofed Motion, and the different Degrees of Celerity of the fimple Motions; provided we know that thofe Motions are equal or uniform to each other. For when we have two Points of a Right Line, we have it entire, and we can compare the Right Line A E, or the compofed Motion that is known, with the Lines A B, and A C, that is, with the fimple Motions that are unknown.
Now let us afreiri luppoie a Stone driven from A to B, by an uniform Motion, but defcending towards E with an unequal, like to that which ponderous Bodies are thought generally to tend to the Centre of the Earth, according to the common Opinion j that is to fay, let the Spaces which it pafles over be amongft themfelves as the Squares of the Times in which it pafles them over; the Line which it fhall defcribe will be a Parabola, and the Point in which the Stone (hall be at every Moment of its Motion, may be determined with the utmoft Nicety and Ex- affnefs.
^ For, if at the firft Moment that Body falls Two Foot from A towards C, in the fecond Six, in che third Chap. IV. The Search after Truth.
third Ten, in the fourth Fourteen, and that it be driven by an uniform Ifaotion from A towards B, which is Sixteen Foot in length •, 'tis evident, that the Line which that Body defcribes is a Para- tola, whofe Parameter is Eight Foot long \ becaufe the Square of the Lines that are applied the Diameter, which Lines mark the Times, and the regular Motion of A towards B,is equal to the Retlangle of the Parameter, through the Lines that mark the unequal and accelerated Motions- lb that the Squares of the applied Lines, or^the Squares of the Times, will be amonglt them- lelves, as the Parts of the Diameter contained betwixt the Pole and the applied Lines.
The bare looking on the fixth Figure is fufficient to perfuade us of all this •, for the Semicircles (hew that A 2 is to A 4, that is, to the applied Line 2 X, its equal, as 2 X is to A 8; That. A 1 8 is to A 1 2, that is, to the applied Line 1 8 X, as 1 8 X is tcf A 8, 0c\ And therefore, that the Rectangles A 2 by A 8, and A 18 by A 8, are equal to the Squares of 2 X and 18 X, &c. andcon- iequently thofe Squares have the fame Proportion to each other, as thofe Reclangles.
The Parallels upon A B and A C, which cut each other at the Points X X X, do alfo fenfibly mew the Way of that Body, and the Places in which it mull be at fuch a time. Lailly, They reprefent to the Eyes the true Degrees of the corripofed Motion, and of its Acceleration, in any determinate Time...Let's fuppofe again a Body moving from A towards B and C, but unequally on both fides. If that Inequality be always and every where alike, or if it either encreafes or diminifhes in the lame proportion, the Line which it fhall defcribe will be a Right.
And though there fhould be an Inequality, either in the Augmentation or Diminution of the' fimple Motions, whatever that Inequality be, it will not be hard to find the Line that reprefents to the Imagination the Motion compofed of the fimple Motions, if you exprefs thofe Motions by Lines, and draw to thefe Lines Parallels cdtting each other: For, the Line that mail pafs through Jill the Interferons of thofe Parallels, will reprefent the Motion compofed of thofe Motions that are unequal, and unequally increafed or diminifhed.
For example, If we fuppofe that a Body is moved by two equal or unequal Forces, whatever they be; that one of thole Motions ftill encreafes, or diminifhes, in any given Geometrical, or Arithmetical Proportion •, and that the other Motion encreafes or diminifhes in fome other Arith- metical, or Geometrical Progreffion 5 to find out the Points through which the Line muft pafs that reprefents to the Eyes and Imagination the Motions compofed of thofe Motions, draw, as has been laid, the two Lines A B and A C, that exprefs the fimple Motions, and divide thofe Lines as thofe leveral Motions are fuppofed to accelerate, at the Points 1, 2, 5, 4, 5: If the Motion reprefented by the Line A C encreafes or diminifhes in fuch an Arithmetical Progreffion as 1, 2, 3, 4, 5. And if the Motion reprefented by the Line A B increafes in this double Progref-. fi'on, 1, 2, 4, 8, 16, or diminifhes in the fubdu- ple Progreffion, 4, 2, 1, f, 4, i, divide it at the Points 1, 2, 4, 8, 16, or 4, 2, 1, i, i, i-, laftly, draw through thofe Divifions Parallels to A B and A C,and the Line AE, that pafles through all the Points of the Interie&ion of thofe Parallels, will be the Line reprefenting the compofed Motion, and the Way through which the Body moves.
If we defire exacfly to know how long a Body has been in coming to fuch a Point, from its fet- ting out; the Parallels drawn from the Point upon A B, or A Q will fhew it -, for the Divifions of A B, and A C, mark the Time. And likewife, if we defire to know the Place to which a Body fhall arrive within fome certain Time, the Parallels drawn from the Divifions of the Lines A B and A C, that reprefent the Time, will, by their Interfe£lion, fhew 11s the Point we feek for. As to its Diftance from the Term whence it has begun to move, it will always be eafie to know it, by drawing a Line from that Point towards A; for the Length of that Line will be known, by comparing it either to A B or AC, which are known. But as to. the Length of the Way through which that Body has run, in advancing to this Point, it will ftiil be hard to disco- ver it; becaufe A E, the Line of its Motion, being crooked, cannot be compared with either of thofe Right Lines.
If you would determine the infinite Points through which that Body rriuft pafs, that is, nicely defcribe, and by a continual Motion, the Line AE, you had need make a Pair of CompafTes that fhould move according to all the Conditions exprefs'd in the Suppositions that have been mentioned; which would be very difficult to invent, and impoffible to perform, and almoft ur> profitable, to difcover the Relations of Things betwixt themfelves •, fince commonly we need not all the Points of which a Line is compofed, but only fome, to help the Imagining Faculty, when it confiders thofe Motions* Thole Inftances are fufficient to fhew, that we may by Lines exprefs and reprefent to the Ima- gination moft of our Ideas; and that Geometry, which teaches to compare thofe Lines, and thereby know their feveral Relations, is of a greater ufe and extent than is commonly fuppofed. For, Aftronomy, Mufick, Mechanicks, and generally all the Sciences, whofe Objects are fufceptible «f more or lefs, and may.be confidefd under the Notion of extended, that is to fay, all accurate Sciences, 43 44 £ M alee ran che Concerning Book YL Sciences, may be referfd to Geometry •, becaufe all Speculative Truths, cbiififtiag in the Relations of things, or in Relations betwixt their Relations, they may all be referr'd to Lines h Geometrical Conl'equences may be drawn, from them; and when thofe Confequences are made fenfible by Lines, 'tis almolt impoflible to miftake. Thus may Sciences be carried very far with great eafinels.
For Inltance, The Jleafon why we diftinctly know, and precifely 'mark an Octave, a Fifth, a Fourth in Mufick, is that the Sounds are exprefled by Strings exactly divided; and that we know that the String which founds an Ocfave is in double proportion with that from whence the O&ave riles-, that a Fifth is with it in a Sefquialter Proportion, or. as 3 to 2, and fo of the reft. For the Ear alone cannot judge of Sounds, with fb much nicety and accuracy as a Science requires. The molt skilful Practitioners, the moft delicate and nicelt Ears are not fenfible enough to ob- ferve the difference betwixt certain Sounds, and judging of things by the Seniation they have of them, fafly imagine that there's none at alh Some cannot diltinguilh betwixt an Octave and 3 thirds, others fancy that the Major Tone differs not from the Minor-, fo that the Comma, which, is their Difference, is infenfible to thenv and much more the Schifma, which is but the half of the Comma. ari And therefore, 'tis Reafori alone that manifeftly fhews us, that the fpace of the String which makes the Difference betwixt certain Sounds, being divifible into feverai parts, there may ftill be a great number of different Sounds, very ufefuli for Mufick, which the Ear cannot diftinguifh. Whence it plainly appears, that without Arithmetick and Geometry, we fhould have no exacf and regular Knowledge of Mufick; neither could we fucceed in that Science but by Chance and Ima- gination, and fo Mufick would ceafe from being a Science, grounded upon undeniable Demonftra- tions. In the mean while it muft be granted, that the Songs which owe their birth to the ftrength of Imagination, are, for the moft part, finer and more pleafant to the Senfes, than thofe that are cornpoled by Rule.
• And like wife in Mechanicks, the Heavinefs of a Body, and the Diftance of the Centre of Hea- vinels from its Prop, being capable of more of lefs, both may be figured by Lines: So tnat Geo- metry is ufefull to difcover and demonftrate an infinite number of new Inventions, very convenient to this Life, andpleafing to the Mind, becaufe of their Evidence.
For Inftance^ If a Weight of fix pounds is to be put in (equilibrium with one of three, let that Weight of fix pound hang on the Arm of a Balance, at two Foot diftance from the Prop: then, only knowing this general Principle of all Mechanicks, That Weights to jland in equilibrium, muft be in a reciprocal proportion with their Diftanees from the Prop; ( that is, That one Weight muft be to the other, as the Diftance betwixt the laft Weight and the Prop is to the Diftance of the firft. Weight from the faid Prop; ) it will be eafie to find out by Geometry, what muft be the Di- ftance of a Weight of Three pounds, that all may remain in aquilibrio; if you find by the Twelfth Proportion of the Sixth Book of Euclid, a fourth proportional Line, which here will be of four! Foot. So that you may plainly difcover all the Truths that depend1 upon that fundamental Prin- ciple of Mechanicks, ( when once known, )\ by the ufe of Geometry j that is* by reprefenting withr Lines whatever can be considered in Mechanicks.
Geometrical Lines and Figures are therefore moft proper to reprefent to the Imagination, the Relations betwixt Magnitudes, or betwixt things that differ in degree of more and leis, as Spaces,1 Times, Weights, &c. as well becaufe they- are moft fimple Objecls, as that they are imagin'ct with great eafinefs, It may even be laid, "to the Honour of Geometry, That Lines can repreleht to the Imagination more things than the Mind can know. Since Lines can exprels the Relations of incommenfurable Magnitudes, that is, fuch Relations as cannot be known, becaufe there is no common Meafure to compare them together. But that Advantage is not very confiderable, as to the Search after Truth ■, becaufe thofe fenfible Representations of incommenfurable Magnitudes, difcover nothing to the Mind.
Geometry is therefore exceedingly ufeful, to make the Mind attentive to thofe things, whole Relations we defire to difcover: However it muft be granted, that it is fometimes an Occafion of Errour, becaafe the evident and pleafant Demonftrations of that Science, takes us up fo much, that we have not a fufficient Regard for the Confideration of Nature. Thence it comes, that the new-invented Engines do not all fucceed L; that thofe Mufical Compofures, in which the Propor- tions of Confonances are belt obferved, are not always the moft grateful; and that the moft ac- curate Calculations of Aflronomy do not always beft foretell the Incidence and Duration of E- clipfes. Nature is not abftracf ed •, Levers and Wheels, in Mechanicks, are not Mathematical Lines and Circles: All Men are not pleafed with the fame Mufical Tunes, nor even the fame Man at different times -, for their Satisfaction proceeds from the Commotions of their Spirits, than which nothing can be more variable. And as to Aflronomy, the Courfe of the Planets is not perfectly regular, whilft floating in the valt Spaces they are irregularly carried by the fluid Matter that fur- rounds them: So that the Errours of AJlronomy, Mujick, Mechanicks, and all Sciences in which