Geometry is ufed, are not to be afcribed to that undoubted Science, but to the falfe Application that is madtof it.
For Inltance, we fuppofe that Planets, by their Motion, defcribe Circles and Ellipfes perfectly regular, And though that be not exa£tly true, yet w» doe well to fuppofe it fo, that we may- draw Inferences horn thence, and becaule it wants but little of being true; but we muft ftill re- member, that the Principle from which we argue is a Suppoftion. Likewife in.Mechanicks, we fuppofe Wheels and levers perfectly hard without gravity and rubbing, and like to Mathemati- cal Lines and Circles: or rather, we have not a futftcient confideration for the faid Gravity and rubbing, Chap. IV. The Search after Truth. 4.5 rubbing, for the Nature of the Matter, and the Relation thofe things have betwixt them. We mind not that Hardnefs and Bulk increafe Heavinefsj Heavinefs fretting, whilft fretting diminifhes Force, and caufes the Engine to break, or wears it out very quickly: So that what often fucceeds upon a fmall portion of Matter, feldom takes effeft upon a great Body.
No wonder therefore if we miftake, fince we argue from Principles not fully known -, nor yet becaufe it rids us not of all Errours, mult we imagine Geometry ufelefs; It makes us draw from our Suppofitions very true and coniequential Inferences; and affords us an evident Knowledge of what we confider, by making us attentive.- We can even difcover by its means, the Failhood of our Suppofitions =, for being certain of the Truth of our Reafonings, which however do not agree with Experience, we difcover that our Principles are falfe. But without Geometry and Aritbme- tick we can difcover nothing, that is fomewhat difficult, in the molt accurate Sciences, though we argue from certain and undeniable Principles.
We muft then look upon Geometry as a fort of univerfal Science, which opens and enlarges the • Mind, makes it attentive, and affords it fo much Skill as to regulate its Imagination, and to draw from it all the poffible Succours. For by the affiftance of Geometry, the Mind regulates the Mo-.
tion of the Imagination, and the Imagination regulated keeps up the View and Application of the Mind.
the Mind.
But that we may learn to make a good life of Geometry, we muft obferve that all the things that fall under the Imagination, are not as eafily imaginable one as the other; fince all the.Ima- ges do not equally fill the Capacity of the Mind. Tis more difficult to imagine a Solid than a Plain, and this than a fimple Line •, becaufe the clear perception of a Solid requires a greater thought than that of a Plain, and a Line. Even Lines differ, as to this, amongft themfelves -, a Parobolick, Elliptick, or fome other very compofed Line, requires more thinking, that is, takes up the Mind, more than the Figure of a Circle, and this than a right Line; becaufe 'tis harder to imagine Lines that are defcribed by very compofed Motions, and have feyeral different Relations, than thofe that are drawn by Motions very fimple, and have but a few Relations. For Relations cannot be clear- ly perceived without the Attention of the Mind to feveral things, and as their number is greater, id muft the thought or the perception be more extended. Hence it happens, that there are Fi- gures fo much compofed, that they extend beyond the reach of a diftincf Imagination -, whereas others may be imagin'd with great facility.
Amongft the three forts of Right-lined Angles, viz. the acute, the right, and the obtufe, none but the Right raifes a very diftinct and determinate Idea. For as there are an Infinity of either acute or obtufe Angles, that differ all from one another; fo we can imagine nothing nicely nor diftin£tly, when we imagine an acute or obtufe Angle. But we cannot be miftaken in imagining a right Angle; the Idea of it is fo very diftincf, and its Image which it raifes in the Brain fo very True it is, that we may determine the general and indefinite Idea of an acute Angle to the par- ticular Idea of an Angle of 30 degrees, which Idea is as accurate as that of an Angle of 90, that is, of a right Angle; but the Image of it, which we may endeavour to imprint on the Brain, will. ne- ver be fo very exaCt, as that of a right Angle •, being not ufed to defcribe that Image, we cannot draw it but by thinking on a Circle, or on the determinate Portion of a Circle divided into equal Parts. But to imagine a right Angle, we need not think on that divifion of a Circle; the bare Idea of a Perpendicular is fufficient for the Imagination to draw the Image of that Angle; and we can reprefent Perpendiculars without trouble, being ufed to fee all things Handing upright.
Hence it is eafie to judge, That to have a fimple, diftincF, and well-determin'd Objeft, apt to be eafily imagin'd, and confequently to make the Mind attentive, and to promote its Evidence in the Truths it is in queft of-, we muft reduce all the Magnitudes we eonfider to plain Superficies, termin'd by Lines and right Angles, as are perfect Squares, and other right Angled Figures, or to bare right Lines; for thefe are the Figures whofe nature is the moft eafily known.
We pretend not however that all the Subjects of our Knowledge and Enquiry, may be repre- fented by Geometrical Lines and Figures. There are many, which neither can nor ought to be brought under that Rule. Forlnftance, the Knowledge of a God, Allmighty, All-juft, on whom all things depend all manner of ways, Who commands his Creatures to obey his Orders, that they may be capable of Happinefs; that Knowledge, I fay, is the Principle of all Morality, and of an infinite number of certain and undoubted Confequences; yet neither the Principle, nor the Confe- quences, can be reprefented by Geometrical Figures. Neither is it poffible to figure and reprefent by Lines many Notions of Natural Philofophy, which yet may evidently difcover to us feveral Truths. However, it may be truly laid, that an Infinity of things may be examin'd and learn'd by that Geometrical Method; which is ever advantageouily imploy'd, fince it accuftoms the Mind to Attention, by caufing it to make a regular ufe of its Imagination; and that things which are > learn'd that way, are more clearly demonftrated, and eafier retain'd than others.
I might have afcribed to the Senfes, the Affiftances we derive from Geometry, to preferve the Attention of the Mind; but though Lines be fomething fenfible, yet, I thought, Geometry be- longs rather to the Imagination, than to the Senfes. It would be unprofitable to fet down my Reafons for it^ which could only juftifie the order I have obferved in this Treatife: and that's a thing not very material to our purpofe. I have not yet fpoken of Arithmetic^ and Algebra, becaufe the Cyphers and Letters of the Alphabet, that are ufed in thofe Sciences, are not fb fer- viceable to ftrengrhen the Attention of the Mind as to encreafe its Extent, as we fhall explain it in the following Chapter.
U ■ Thefe q.6 F. Male branch e Concerning BookVt.
Thefe are the general Helps to improve the Attention of the Mind: I know of no other, be- fides a firm Refolution of being attentive •, of which we forbear to fpeak, becaufe we fuppofe it in thofe that give up themfelves to ftudy.
There are, however, fome others particular to fome Perfons •, as, fome Meats, fbme Drinks^, Fome Places, fome Difpofitions of the Body, and the like; which every one muft learn from Ex- perience, oblerving the State of his Imagination after the Meal, and what Things belt prelerve, or moil diffipate the Attention of the Mind. This only may be laid in general, That the moderate Life of fuch Aliments as make many Animal Spirits, is very fit to improve the Attention of the Mind, and the Strength of the Imagination, when 'tis weak and languifhing.
CHAP. V.
Of the Means to improve the Tixtent and Capacity of the Mind: That Arith- metick and Algebra are of abfolute NeceJJity to it.
W' E ought not haftily to imagine, that the Extent and Capacity of the Mind can really be increafed. The Humane Soul is, if I may fo fpeak, a determined Quantity, or a Portion of Thought, contained within fome certain Bounds, which me cannot pafs: She cannot grow greater, or more capacious than (he is: She neither fwells up, nor dilates, as 'tis commonly be: lieved of Liquors and Metals 3 and perceives never more at one time, than another.
This, 1 confefs, feems contrary to Experience; fince fometimes we think upon many Objects, and fometimes but upon one; and even we often fuppofe, that we think upon nothing. How- ever,, if it be confider'd, thatThonght is to the Soul, what Extenfion is to Matter •, it will plain- ly appear, that as a Body cannot truly be more extended at one time than another, fo, if we conceive it right, the Soul cannot think more at one time than another; whether it be then that me perceives many Objects, or is taken up with one, or even when fhe is faid to think upon nothing.
But the Reafon why we imagine that we think more at one time than another, is, that we do not fufficiently diftinguifh betwixt confufed and diftinct Perceptions. More Thought is doubtlefs required, or the Capacity of Thinking muft be more fill'd, diftin&ly to perceive feveral Objefts, than one alone •, but we need not more Thought, to perceive many Things confufedly, than one alone difiintfly. Thus the Degrees or Quantity of Thought is equal in the Soul, when ihe con- fiders many Things, and when fhe confiders only one: For, when (he is taken up with one Thing, fhe has always a clearer Idea of it, than when (lie applies her felf to many.
For, 'tis fit to be obferved, That a fimple Perception fometimes contains as much Thought, or fills as much the Thinking Capacity of the Mind, as a Judgment, and even a compofed Reafon- ing •, fince Experience teaches us, that the fimple but lively, clear, and evident Perception of one Thing, engages our Application, and polTeiTes us as much, as a compofed Reafoning, or tjie ob- fcure and confufed Perception of feveral Relations betwixt many Things.
For, as there is as much or more Senfation in the fenfible Sight of an Obje£t, which I hold near my Eyes, and curioufly examine 3 than in the Sight of a fpacious Field, on which I caft a negligent and carelefs Eye 3 becaufe the nearnefs of the Senfation of the Object near my Eyes, makes up for the Extent of that confufed Senlation of thole many Things which I flightly and unatten- tively look upon in a Field: So the fpiritual Sight the Mind hath of an Object, is often fo lively and diftinci, that it contains as much and more Thought, than the View of the Relations betwixt, many Things.
True it is, that at fome certain times it feems to us as though we thought but upon one Thing, which yet we can hardly comprehend ^ whereas at other times we comprehend that Thing, and feveral others, with great eafinefs: Thence we imagine, that the Soul has more Extent, and a lar- ger Capacity of Thinking, at one time, than at another. But our Miftake is vifible; for the Rea- fon why at fome certain times we can fcarce conceive the eafieft Things, proceeds not from the Capacity of the Soul's being ftraitned or impair'd 3 but from its being fill'd with fome lively Senfation of Pain or Pleafure, or with a great number of weak and dark Seniations, that caufe a fort of Giddinefs, which is commonly nothing elfe but the confufed Senfation of a great num- ber of Things.
A Piece of Wax is fufceptible of a very diftinci: Figure; but cannot admit two, without a Mixture of both-, fince it cannot be perfectly round and fquare at the fame time: and if one fhould pretend to give it a Million of Figures, none of them would be diftinci:. And in that Cafe, fuppofing that Piece of Wax capable of knowing its own Figures, yet it could not tell which it is that terminates it on all fides, the number would be fo great. It is even fo with our Soul, when a very great number of Modifications take up her Capacity; (lie can perceive none diitinctly, becaufe fhe has not a feparate Senfation of them, and fo thinks file is fenfible of no- thing. She cannot fay that (he feels Pain, Pleafure, Light, Sound, Savour; 'tis none of thofe Qualities, and yec 'tis them all together, fhe is fenfible of And though we fhould fuppofe that the Soul is not f lib] eft to the confufed and unruly Motion pi 1 he Animal Spirits, and fo five from the Contagion of 'her Body, as to have her Thoughts al- together Chap. V. The Search after Truth. 'a% together independent on what happens in it; yet it might fall out that we fhould eafier under- ftand fome Things at one time than at another, without any Enlargement or Diminution in the Capacity of" our Soul ■, for then we might think upon particular Objects, or of Being indefinite and in general.
The general Idea of Infinite is infeparable from the Mind, and wholly takes up its Capacity, whenever it thinks upon no particular Thing: For when we lay, that we think on nothing, it fig- nifies not that we think not upon that general Idea, but only that our Thoughts are not applied to any particular Objecf.
And certainly, if that Idea did not fill our Mind, we could not think, as we do, upon all forts of Things ^ fince we cannot think upon Objefts of which we have no Knowledge. And if thai- Idea were not moreprefent to the Mind when we fuppofe we think upon nothing, than when we are bufie about fome particular Object, we could as eafily think upon whatever we pleafe, when we are mightily taken up with fome particular Truth, as when we are not" attentive unto any thing: Which is repugnant to Experience. For, to inftance, when we are ftrongly erigag'd in meditating on fome Geometrical Propofition, we find not lb much eafinefs to think upon other Things, as when we are diverted by no particular Thought. And therefore we think more on the General and Infinite Being, when we think lefs on the Particular and Finite -, and we think al- ways as much at one time as at another.
We cannot then improve the Extent and Capacity of the Mind, by fwelling it up, as I may fay, and giving it more Reality than it has received from Nature: But only by a skilful and dex- terous managing thereof -, which is done to the bell advantage by Arithmetick and Algebra: For thofe Sciences afford Means of abridging Ideas fo methodically, and reducing them into fuch an Order, as that the Mind, with its little Extent, is capable, with their Afliltance, of difcovering very compofed Truths, and fuch as appear at nrft fight incomprehenfible. But we muff diaw thefe Things from their Principle, that we may explain them with more clearnefs and certainty.
Truth is nothing elfe but a real Relation, either of Equality or Inequality: Whereas Falfehood is but the Negation of Truth, or a falfe and fantaftick Relation. Truth is that vohkh is, and Falfe- hood is not; or, if you will, is that which is not. We never milfake when we fee Relations that are, fmce we cannot be deceived when we fee the Truth t But we always miftake, when we judge that we fee fome Relations that are not in being; for then we fee a Falfehood, we fee what is not, or rather we fee not at all. Whoever fees a Relation of Equality betwixt two times Two, and Four, fees a Truth, becaufe there is fuch a Relation as he fees; and whoever fees a Relation of Inequality betwixt twice Two, and Five, fees a Truth, becaufe he fees a Relation that really is: But whoever judges that he fees a Relation of Equality betwixt two times Two, and Five, miftakes, becaufe he fees, or rather fuppofes he fees, a Relation of Equality where there is none. Truths are but Relations, and the Knowledge of Truths is the Knowledge of Relations: But Falfehood is not, and the Knowledge of Falfehood, or a falfe Knowledge, is, if it may be fo laid, the Knowledge of what is not; and what is not, cannot be known, but by Relation to what is: So Errour cannot be underftood, but by comparing it to Truth.
There may be diftinguifhed as many Species of Falfehood, as of Truth; and as there are Three forts of Relations, viz, of one Idea to another •, of an Objeft to its Idea, or of an Idea to its Objecf •, and laftly, of one Objecf to another: So there are Three kinds of Truth and Falfehood h namely, betwixt Ideas, betwixt Things and their Ideas, and betwixt Things themfelves. It is true, that 2 times 2 are 4; 'tis falfe, that twice 2 are 5: That is a Truth and a Falfehood betwixt Ideas. 3Tis true, that there is one Sun; 'tis falfe, that there are two 1 Here you have a Truth and a Falfe- hood betwixt Things and their Ideas. 'Tis true, that the Earth is bigger than the Moon -, and 'tis falfe, that rite Sun is fmaller than the Earth: There is a Truth and Falfehood betwixt Objefts themfelves.
Of thofe Three forts of Truths, fuch as are betwixt Ideas are Eternal and Immutable, and upon that account are the Rule andMeafure of all others ^ becaufe every Rule and Meafure ought to be unchangeable. And as Arithmetick, Algebra, and Geometry, are general Sciences, that rule and contain all the particular; fo they only confider thofe forts of Truths. All Truths or Relations betwixt Creatures, or betwixt Ideas and created Things, are obnoxious to thofe Changes whereof Creatures are fufceptible. Nothing but the Truth betwixt our Ideas and the Sovereign Being, or betwixt Ideas themfelves, is Immutable; becaufe neither God, nor the Ideas he contains, are fubjeft. to Alteration.
And therefore 'tis only that fort of Truths which are betwixt our Ideas, that we try to difco- ver by the Exercife of our Reafon 3 fince we, for the molf part, make ufe of our Senfes to difco- ver the others •, as,- we ufe our Eyes and Hands to afcertain us of the Exiftence of Things, and to know the Relations of Equality or Inequality betwixt them. There is nothing but Ideas of which * the Mind can Infallibly know the Relations by it felf, and without the ufe of Senfes. But there are not only Relations betwixt our Ideas, there are alfo Relations betwixt the Relations of our Ideas, betwixt the Relations of thofe Relations, betwixt the Collection of many Relations, and fo ad infinitum; that is to fay, that there are Truths infinitely compounded and perplexed. In Geometrical Stile we call a fimple Truth, or the Relation of one Idea to another, (as the Relation of 4 to 2, or to 2 times 2) a Geometrical Reafon, or only a Reafon: For the Excels and Defect of an Idea, or, to ufe the common Terms, the Excefs or Defe£t of a Magnitude, is not properly a Reafon; nor equal Exceffes and Defects, equal Reafons. When the Ideas or Mag- nitudes are equal, there is a Reafon of Equality, and one of Inequality when they are unequal.
The The 48 F. Malebranche Concerning BdokVI.
The Relation betwixt Relations of Magnitudes, that is to fay, between Reafons, is called Com- pounded Reafon, becaufe 'tis a compounded Relation; as the Relation of 6 to 4, and 3 to 2. When the Compounding Reafons are equal, the Compounded bears the Name of Proportion, or Duplicate Reafon. The Relations of 8 to 4, and 6 to 3, are a Proportion; becaufe thole two Relations are equal.
It mult be obferved, That all the Relations or Reafons, as well fimple as compounded, are true Magnitudes, that very Name of Magnitude being a relative Term, and neceflarily importing a Relation: For, there is nothing Great by it felf, and without Relation to another, belides the Infinite or Unite. All entire Numbers are as true Relations as Fractions themfelves, or as Num- bers compared with, or divided by, others • though we do not confider this, becaufe entire Num- bers may be exprefs d by one Arithmetical Figure. So 4, for inftance, or 1, is as true a Relation as v, or s, though the finite to which 4 relates, be not expreffed, but underftood, 4 being equal to ?, or -*- -,, and therefore every Magnitude being a Relation, or every Relation being a Magnitude, it is plain that we can exprefs all Magnitudes by Cyphers, and reprefent them by Signs to the Imagination.
So that all Truths being but Relations, to know all Truths exactly, both fimple and compound- ed, it is fufficient to have an exact Knowledge of all forts of Relations, fimple and compound: We have already obferved, that there are two, viz. Relations of Equality, and Inequality. It is plain, thatnhoie of Equality are alike; and that as foon as we know that a Thing is equal to another that is known, we have an accurate Knowledge of its Relation: But it goes not fo with Inequality-, fbrbecauie we know that a Tower is higher than a Fathom, and lower than a thou- fand, it follows not that we have a true Idea of its Heighth, or of its Relation with a Fathom.
To compare things together, or rather critically to meafure the Relations of Inequality, there is required a very exact Meafure, a fimple and very intelligible Idea, an univerial Meafure, which may be adapted to all forts of Subjects. That Meafure is Unity, which ferves to meafure all Things, and without which 'tis impoflible to have an accurate Knowledge of any. But all Num- bers being made up of Unites, 'tis evident, that without the Ideas of Numbers, and a Method of comparing and m^afuring rhofe Ideas, that is, without Aritbmetick, 'tis not poflible to make any Progrefs in the Knowledge of Compound Truths.
And as Ideas, and the Relations betwixt Ideas; in fhoit, all lorts of Magnitudes can be great- er or lefs than others -, fo they cannot be made equal, but by more or lefs Unites join'd, or re- peated as often as 'tis neceffary: So that it is only by the Addition and Subtra£tion of Unity, or ot the Parts of the Unity (when 'tis conceived as divided) that we exactly meafure all forts of Magnitudes, and difcover all lorts of Truths. Now Aritbmetick and Algebra are, of all Sci- ences, thofe that afford us molt Skill and Light to effect thole Operations, and to manage the Ca- pacity of the Mind to the belt Advantage, fince they endue it with all the Perfe£tion and Extent that it is capable of, and teach it to dilcover all the Truths that can be exactly known.
For ordinary Geometry does not fo perfe£t the Mind, as the Imagination; and the Truths which that Science difcovers, are not always fo evident as the Matters of it fanfie. For inftance: They fuppofe they have exprefs'd the Value of fome Magnitudes, when they have proved them to be equal to lome Lines, that are the Subtendant of Right Angles, whofe Sides are perfectly known • or to others, that are determined by fome one of the Conick Sections. But their Miltake is vi- fible; for thofe Subtendants are unknown themfelves. We know more exaftly the V 8, or the V 26, than a Line imagined or deferibed upon Paper, to be the Subtendant of a Right Angle, whofe Sides are 2, or one Side of which is 2, and the other 4: At leaft we know, that the r 4 is very near 3, and that the V 20 is about 4 \ -, and there are Rules to come infinitely nearer and nearer the true Magnitude; and if we cannot attain to it, 'tis becaufe the Mind cannot compre-' hend Infinite. Whereas we have but a very confufed Idea of the Magnitude of Subtendant Lines, and are even obliged to have recourfe to the V 8, or the V 20, to exprefs them. So that the Geometrical Conftructions that are ufed to reprefent the Value of unknown Quantities, are not fo conducible to the Mind, to dilcover the Relations or Truths fought for, as to rule the Imagi- nation: But as we are more inclined to imploy our Imagination, than our Mind; fo Men of Learning have commonly more elteem for Geometry, than for Aritbmetick and Algebra.
To underltand perfectly, that Aritbmetick and Algebra, join'd together, are a real Logick, or the Means to difcover the Truth, and afford the Mind as much Extent as it can acquire, it is fuffi* cient to make lome Reflections upon the Rules of thole Sciences.
We have obferved, That all Truths are but Relations; that the molt fimple, and belt known of all, is that of Equality -, that it is the initial Relation, from whence we mult begin to meafure • others, whereby to have an exact Idea of Inequality -, that the Meafure of Inequality is the Unite, which mult be repeated or fubtracted as often as the Excefs or Defect of unequal Magnitudes require it.
Thence it is plain, that all the Operations that may be fubfervient to difcover the Relations of Equality, are only Additions and Subtractions •, Additions ot Magnitudes, to make Magnitudes even; Additions of Relations, to make equal Relations, or to put Magnitudes in proportion with each other •, and laltly, Additions of the Relations of Relations, to equal Relations of Re- lations, or to put Magnitudes in a Compound Proportion.
To equal 4 to 2, we need only add 2 to 2, or fubtract 2 from 4 ^ or laltly, to add the Unite to 2, and fubtract it from 4 •, that's plain. To even the Relation or Reafon oi' 8 to 2, to that of 6 to 3, we mult not add 3 to 2, or fubtract Chap. V. Ihe Search after Truth. ac?
fubtraft 3 from 8, fo that the Excefs of one Number to the other ever mould be equal to?4 -•which is the Excels of 6 above 3 -, that would be an Addition, and Evening of fimple Magni- tudes: But we mull conlider firlt, which is the Magnitude of the Relation of 8 to 2, or what is the Value of | •, and we fhall find, that dividing 8 by 2, the Quotient of that Reafon will be 4, or that | is equal to 4. We mult likewife fee which is the Magnitude of the Relation of 6 to 3; and rinding it equal to 2, we (hall difcover, that thole two Reafons, | equal to 4, and *- equal to 2, differ only by 2: So to make them even, we. may either" add ~ to j-, equal to 2, which will make -~, that is, a Relation equal to 4 •■, or fubtract |, equal to 2, from |, which will make 4, that is, a Relation equal to 4 j or laftly, adding the Unite to -3, and fu trailing it from -*, we fhall have -* and 4, which are equal Relations; for 9 is to 3, as 6 to 2.