Understanding cannot reach. And as it often happens in the Investigation of Truth, especially where it lies at a confiderable Distance from first Principles, that the Number of Connections and Relations are so great, as not to be taken in at once by the most improved Understanding; it is therefore one great Branch of the Art of Invention, to take account of these Relations as they come into View, and dispose of them in such manner, that they may always lie open to the Inspection of the Mind, when disposed to turn its Attention that Way. By this Means, without perplexing ourselves with too many Confiderations at once, we have yet these Relations at Command, when necessary to be taken notice of in the Profecution of our Difcoveries: and the Understanding thus free and difengaged, can bend its Powers more intensely, towards that particular Part of the Investigation it is at present con cerned with. Now in this, according to my Apprehenfion, lies the great Art of human Knowledge; to manage with Skill the Capacity of the Intellect, and contrive fuch Helps, as may bring the most wide and extended Objects within the Compass of its natural Powers. When therefore the Multitude of Relations increase very fast upon us, and grow too unwieldy to be dealt with in the Lump; we must combine them in different Classes, and so dispose of the several Parts, as that they may at all times lie open to the leisurely Survey of the Mind. By this means we avoid Perplexity and Confufion, and are enabled to conduct our Researches, without being puzzled with that infinite Crowd of Particulars, that frequently fall under our Notice in long and difficult Investigations. For by carrying our Attention successively from one Part to another, we can upon occafion take in the Whole; and knowing also the Order and Disposition of the Parts, may have recourse to any of them at Pleasure, when its Aid becomes neceffary in the Course of our Enquiries. great An orderly Difpofition of adapting Objects to the Capacity of fanding. the Under VII. FIRST then I say, than an orderly Combination of Things, and claffing them together with Art and Address, brings great and otherwise unmanageable Objects, upon a level with the Powers of the Mind. We have seen in the first Part of this Treatife, how by taking Numbers in a progreffive Series, and according to an uniform Law of Composition, the most bulky and formidable Collections are comprehended with Ease, and leave diftinct Impreffions in the Understanding. For the several Stages of the Progreffion, serve as so many Steps to the Mind, by which it afcends gradually to the highest Combitions; and as 1. ! as it can carry its Views from one to another with great Eafe and Expedition, it is thence enabled to run over all the Parts feparately, and thereby rise to a just Conception of the Whole. The fame thing happens in all our other complex Notions, especially when they grow very large and complicated; for then it is that we become sensible of the Neceffity of establishing a certain Order and Gradation in the Manner of combining the Parts. This has been already explained at fome Length in the Chapter of the Compofition and Resolution of our Ideas, where we have traced the gradual Progress of the Mind through all the different Orders of Perception, and shewn, that the most expeditious Way of arriving at a just Knowledge of the more compounded Notices of the Underftanding, is by advancing regularly through all the intermediate Steps. Hence it is easy to perceive, what Advantages must arise from a like Conduct, in regard to those several Relations and Connections, upon which the Investigation of Truth depends. For as by this means we are enabled to bring them all within the Reach of the Mind, they can each in their Turns be made use of upon occafion, and furnish their Affiftance towards the Discovery of what we are in quest of. Now this is of principal Confideration in the Bufiness of Invention, to have our Thoughts so much under Command, that in comparing Things together, in order to difcover the Result of their mutual Connections and Dependence, all the feveral Lights that tend to the clearing the Subject we are upon, may lie distinctly open to the Understanding, fo as nothing material shall escape its View: because an Oversight of this Kind in summing up the Account, muft not only greatly retard its Advances, but in many Cafes check its Progress altogether. And in enabling us to procced gradually and wich Eafe in tion of Truth. as we VIII. BUT secondly, another Advantage arising from this orderly Disposition is, that hereby we free the Mind from all unneceflary Fatigue, and leave it to fix its Attention upon any Part the Investiga- feparately, without perplexing itself with the Confideration of the Whole. Unknown Truths, have already observed, are only to be traced, by means of the Relation between them and others that are known. When therefore these Relations become very numerous, it must needs greatly distract the Mind, were it to have its Attention continually upon the Stretch, after fuch a Multitude of Particulars at once. But now, by the Method of claffing and ordering our Perceptions above explained, this Inconvenience is wholly prevented. For a juft 157 Distribution of Things, as it ascertains distinctly the Place of each, enables us to call any of them into view at pleafure, when the prefent Confideration of it becomes neceffary. Hence the Mind, proceeding gradually through the feveral Relations of its Ideas, and marking the Results of them at every Step, can always proportion its Enquiries to its Strength; and confining itself to such a Number of Objects, as it can take in and manage at ease, sees more diftinctly all the Consequences that arife from comparing them one with another. When therefore it comes afterwards to take a Review of these its several Advances, as by this means the Amount of every Step of the Investigation is fairly laid open to its Inspection, by adjusting and putting these together in due Order and Method, it is enabled at last to difcern the Result of the Whole. And thus, as before in the Compofition of our Ideas, so likewise here in the Search and Difcovery of Truth, we are fain to proceed gradually, and by a Series of fucceffive Stages. For these are so many RestingPlaces to the Mind, whence to look about it, survey the Conclufions it has already gained, and fee what Helps they afford, towards the obtaining of others which it must still pass through, before it reaches the End of the Investigation. Hence it often happens, that very remote and distant Truths, which lie far beyond the Reach of any single Effort of the Mind, are yet by this progreffive Method successfully brought to Light, and that too with less Fatigue to the Understanding, than could at first have well been imagined. For although the whole Process taken together, is frequently much too large to come within the View of the Mind at once; and therefore confidered in that Light, may be faid truly to exceed its Grasp: yet the several Steps of the Investigation by themselves, are often easy and manageable enough; so that by proceeding gradually from one to another, and thoroughly maftering the Parts as we advance, we carry on our Researches with wonderous Dispatch, and are at length conducted to that very Truth, with a View to the Discovery of which, the Inquisition itself was set on foot. Aigebra and IX. BUT now perhaps it may not be im- fally ۱ fally known to be the very Art and Principle of Invention; and in Arithmetic too, we are frequently put upon the finding out of unknown Numbers, by means of their Relations and Connections with others that are known; as where it is required to find a Number, equal to the Sum of two others, or the Product of two others. I chuse to borrow my Examples chiefly from the last Science, both because they will be more within the Reach of those, for whom this Treatise is principally designed; as likewife, because Arithmetic furnishes the best Models of a happy Sagacity and Management, in claffing and regulating our Perceptions. So that here more than in any other Branch of human Knowledge, we shall have an Opportunity of observing, how much an orderly Difpofition of Things, tends to the Ease and Success of our Enquiries, by leaving us to canvass the Parts feparately, and thereby rise to a gradual Conception of the Whole, without entangling ourselves with too many Confiderations at once, in any fingle Step of the Investigation. For it will indeed be found, that a Dexterity and Address in the Use of this last Advantage, ferves to facilitate and promote our Discoveries, almost beyond Imagination or Belief. The Metbod of Clafling our Perceptions in Arithmetic. X. We have already explained the Manner of reducing Numbers into Classes, and of diftinguishing these Classes by their several Names. And now we are farther to observe, that the present Method of Notation is so contrived, as exactly to fall in with the Form of numbering. For as in the Names of Numbers, we rise from Units to Tens, from Tens to Hundreds, from Hundreds to Thousands, &c. so likewife in their Notation, the fame Figures, in different Places, fignify these several Combinations. Thus 2 in the first Place on the right Hand denotes two Units, in the second Place it expresses fo many Tens, in the third Hundreds, in the fourth Thousands. By this means it happens, that when a Number is written down in Figures, as every Figure in it expresses some diftinct Combination, and all these Combinations together make up the total Sum; so may the feveral Figures be confidered as the constituent Parts of the Number. Thus the Number 2436 is evidently by the very Notation diftinguished into four Parts, mark'd by the four Figures that ferve to express it. For the first denotes two Thousand, the fecond four Hundred, the third Thirty or three Tens, and the fourth Six. These several Parts, though they here appear in a conjoined Form, may yet be also expreffed feparately thus, 2000, 400, 30 and 6, and the Account is exactly the fame. XI. THIS The Helps thence derived towards an ey Adsition of Numbers. XI. THIS then being the Cafe, if it is required to find a Number, equal to the Sum of two others given; our Business is, to examine separately these given Numbers; and if they appear too large and bulky to be dealt with by a fingle Effort of Thought, then, fince the very Notation diftinguishes them into different Parts, we must content ourselves with confidering the Parts asunder, and finding their Sums one after another. For since the Whole is equal to all its Parts, if we find the Sums of the several Parts of which any two Numbers confift, we certainly find the total Sum of the two Numbers. And therefore, these different Sums, united and put together, according to the established Rules of Notation, will be the very Number we are in quest of. Let it be propofed, for instance, to find a Number equal to the Sum of these two: 2436 and 4352. As the finding of this by a fingle Effort of Thought, would be too violent an Exercise for the Mind; I confider the Figures reprefenting these Numbers, as the Parts of which they confift, and therefore fet myself to discover their Sums one after another. Thus 2 the firft Figure on the right Hand of the one, added to 6 the first Figure on the right Hand of the other, makes 8, which is therefore the Sum of these two Parts. Again, the Sum of 5 and 3, the two Figures or Parts in the second Place, is likewife 8. But now as Figures in the second Place, denote not simple Units, but Tens; hence it is plain, that 5 and 3 here, signify five Tens and three Tens, or 50 and 30, whose Sum therefore must be eight Tens, or 80. And here again I call to mind, that having already obtained one Figure of the Sum, if I place that now found immediately after it, it will thereby stand also in the fecond Place, and fo really express, as it ought to do, eight Tens, or 80. And thus it is happily contrived, that though in the Addition of Tens, I confider the Figures compofing them as denoting only simple Units, which makes the Operation easier, and lefs perplexed; yet by the Place their Sum obtains in the Number found, it expreffes the real Amount of the Parts added, taken in their full and compleat Values. The fame thing happens in fumming the Hundreds and Thousands; that is, though the Figures expreffing these Combinations, are added together as fimple Units, yet their Sums standing in the third and fourth Places of the Number found, thereby really denote Hundreds and Theufands, and fo reprefent the true Value of the Parts added. XII. HERE |