ANALYSIS
, is, generally, the resolution of any thing into its component parts, to discover the thing or the composition. And in mathematics it is properly the method of resolving problems, by reducing them to equations. Analysis may be distinguished into the ancient and the modern.
The ancient analysis, as described by Pappus, is the method of proceeding from the thing sought as taken for granted, through its consequences, to something that is really granted or known; in which sense it is the reverse of synthesis or composition, in which we lay that down first which was the last step of the analysis, and tracing the steps of the analysis back, making that antecedent here which was consequent there, till we arrive at the thing sought, which was taken or assumed as granted in the first step of the analysis. This chiefly respected geometrical enquiries.
The principal authors on the ancient analysis, as recounted by Pappus, in the 7th book of his Mathematical Collections, are Euclid in his Data, Porismata, & de Locis ad Superficiem; Apollonius de Sectione Rationis, de Sectione Spatii, de Taclionibus, de Inclinationibus, de Locis Planis, & de Sectionibus Conicis; Aristæus, de Locis Solidis; and Eratosthenes, de Mediis Proportionalibus; from which Pappus gives many examples in the same book. T these authors we may add Pappus himself. The same sort of analysis has also been well cultivated by many of the moderns; as Fermat, Viviani, Getaldus, Snellius, Huygens, Simson, Stewart, Lawson, &c, and more especially Hugo d'Omerique, in his Analysis Geometrica, in which he has endeavoured to restore the Analysis of the ancients. And, on this head, Dr. Pemberton tells us “that Sir Ifaac Newton used to censure himself for not following the ancients more closely than he did; and spoke with regret of his mistake, at the beginning of his mathematical studies, in applying himself to the works of Descartes, and other algebraical writers, before he had considered the Elements of Euclid with that attention so excellent a writer deserves: that he highly approved the laudable attempt of Hugo d'Omerique to restore the ancient analysis.”
In the application of the ancient analysis in geometrical problems, every thing cannot be brought within strict rules; nor any invariable directions given, by which we may succeed in all cases; but some previous preparation is necessary, a kind of mental contrivance and construction, to form a connexion between the data and quæsita, which must be left to every one's fancy to find out; being various, according to the various nature of the problems proposed: Right lines must be drawn in particular directions, or of particular magnitudes; bisecting perhaps a given angle, or perpendicular to a given line; or perhaps tangents must be drawn to a given curve, from a given point; or circles described from a given centre, with a given radius, or touching given lines, or other given circles; or such-like other operations. Whoever is conversant with the works of Archimedes, Apollonius, or Pappus, well knows that they founded their analysis upon some such previous operations; and the great skill of the analyst consists in discovering the most proper affections on which to found his analysis: for the same problem may often be effected in many different ways: of which it may be proper to give here an example or two. Let there be taken, for instance, this problem, which is the 155th prop. of the 7th book of Pappus.
From the extremities of the base A, B, of a given segment of a circle, it is required to draw two lines AC, BC, meeting at a point C in the circumference, so that they shall have a given ratio to each other, suppose that of F to G.
The solution of this problem, as given by Pappus, is thus.
Suppose the thing done, and that the point C is found: then suppose CD is drawn a tangent to the circle at C, and meeting the line AB produced in the point D. Now by the hypothesis AC : BC :: F : G, and also AC^{2} : BC^{2} :: DA : DB, as may be thus proved.
Since DC touches the circle, and BC cuts it, the angle BCD is equal to BAC by Euc. iii. 32; also the angle | D is common to both the triangles DCA, DCB; these are therefore similar, and so, by vi 4, DA : DC :: DC : DB, and hence DA^{2} : DC^{2} :: DA : DB by cor. vi 20. But also, by vi 4, DA : AC :: DC : CB, and by permutation DA : DC :: AC : BC, or DA^{2} : DC^{2} :: AC^{2} : BC^{2}; and hence, by equality, AC^{2} : BC^{2} :: DA : DB.
But the ratio of AC^{2} to BC^{2} is given by prop. LVII of Simson's edition of the Data, because the ratio of AC to BC is given, and consequently that of DA to DB is given. Now since the ratio of DA to DB is given, therefore also, by Data vi, that of DA to AB, and hence, by Data ii, DA is given in magnitude.
And here the analysis properly ends. For it having been shewn that DA is given, or that a point D may be found in AB produced, such, the a tangent being drawn from it to the circumserence, the point of contact will be the point sought; we may now begin the composition, or synthetical demonstration; which must be done by finding the point D, or laying down the line AD, which, it was affirmed, was given, in the last step of the analysis.
Construction. Make as F^{3} : G^{2} :: AD : DB, (which may be done, since AB is given, by making it as F^{2}G^{2} : G^{2} :: AB : DB, and then by composition it will be as F^{2} : G^{2} :: AD : DB); and then from the point D, thus found, draw a tangent to the circle, and from the point of contact C drawing CA and CB, the thing is done.
Demonstration. Since, by the constr. F^{2} : G^{2} :: AD : DB, and also AD : DB :: AC^{2} : BC^{2}, which has been already demonstrated in the analysis, and might be here proved in the same manner. Therefore F^{2} : G^{2} :: AC^{2} : BC^{2}, and consequently F : G :: AC : BC. Q.E.D.
Here we see an instance of the method of resolution and composition, as it was practised by the ancients, the solution here given being that of Pappus himself. But as the method of referring and reducing every thing to the Data, and constantly quoting the same, may appear now to be tedious and troublesome: and indeed it is unnecessary to those who have already made themselves masters of the substance of that valuable book of Euclid, and have by practice and experience acquired a facility of reasoning in such matters: I shall therefore now shew how we may abate something of the rigour and strict from of the ancient method of solution, without diminishing any part of its admirable elegance and perspicuity. And this may be done by the instance of another solution, of the many more which might be given, of the same problem, as follows.
Let us again suppose that the thing is done, viz AC : BC :: F : G, and let there be drawn BH making the angle ABH equal to the angle ACB, and meeting AC produced in H. Then, the angle A being also common, the two triangles ABC and ABH are equiangular, and therefore, by vi 4, AC : BC :: AB : BH, in a given ratio; and, AB being given, therefore BH is given in position and magnitude.
Construction. Draw BH making the angle ABH equal to that which may be contained in the given segment, and take AB to BH in the given ratio of F to G. Draw ACH, and BC.
Demonstration. The triangles ABC, ABH are equiangular, therefore, vi 4, AC : CB :: AB : BH, which is the given ratio by construction.
Modern Analysis, consists chiefly of algebra, arithmetic of insinites, insinite series, increments, fluxions, &c; of each of which a particular account may be seen under their respective articles.
These form a kind of arithmetical and symbolical analysis, depending partly on modes of arithmetical computation, partly on rules peculiar to the symbols made use of, and partly on rules drawn from the nature and species of the quantities they represent, or from the modes of their existence or generation.
The modern analysis is a general instrument by which the sinest inventions and the greatest improvements have been made in mathematics and philosophy, for near two centuries past. It furnishes the most perfect examples of the manner in which the art of reasoning should be employed; it gives to the mind a wonderful skill for discovering things unknown, by means of a small number that are given; and by employing short and easy symbols for expressing ideas, it presents to the understanding things which otherwise would seem to lie above its sphere. By this means geometrical demonstrations may be greatly abridged: a long train of arguments, in which the mind cannot, without the greatest effort of attention, discover the connection of ideas, is converted into visible symbols; and the various operations which they require, are simply effected by the combination of those symbols. And, what is still more extraordinary, by this artifice, a great number of truths are often expressed in one line only: instead of which, by following the ordinary way of explanation and demonstration, the same truths would occupy whole pages or volumes. And thus, by the bare contemplation of one line of calculation, we may undersland in a short time whole sciences, which otherwise could hardly be comprehended in several years.
It is true that Newton, who best knew all the advantages of analysis in geometry and other sciences, laments, in several parts of his works, that the study of the ancient geometry is abandoned or neglected. And indeed the method employed by the ancients in their geometrical writings, is commonly regarded as more rigorous, than that of the modern analysis: and though it be greatly inferior to that of the moderns, in point of dispatch and facility of invention; it is nevertheless highly useful in strengthening the mind, improving the reasoning faculties, and in accustoming the young mathematician to a pure, clear, and accurate mode of investigation and demonstration, though by a long and laboured process, which he would with difficulty have submitted to if his taste had before been vitiated, as it were, by the more piquant sweets of the modern analysis. And it is principally on this that the complaints of Newton are founded, who feared lest by the too early and frequent use of the modern analysis, the science of geometry should lose that rigour and purity which characterise its investigations, and the mind become debilitated by the | facility of our analysis. This great man was therefore well founded, in recommending, to a certain extent, the study of the ancient geometricians: for, their demonstrations being more difficult, give more exercise to the mind, accustom it to a closer application, give it a greater scope, and habituate it to patience and resolution, so necessary for making discoveries. But this is the only or principal advantage from it; for if we should look no farther than the method of the ancients, it is probable that, even with the best genius, we should have made but few or small discoveries, in comparison of those obtained by means of the modern analysis. And even with regard to the advantage given to investigations made in the manner of the ancients, namely of being more rigorous, it may perhaps be doubted whether this pretension be well founded. For to instance in those of Newton himself, although his demonstrations be managed in the manner of the ancients; yet at the same time it is evident that he investigates his theorems by a method different from that employed in the demonstrations, which are commonly analytical calculations, disguised by substituting the name of lines for their algebraical value: and though it be true that his demonstrations are rigorous, it is no less so that they would be the same when translated and delivered in algebraic language; and what difference can it make in this respect, whether we call a line AB, or denote it by the algebraic character a? Indeed this last designation has this peculiarity, that when all the lines are denoted by algebraic characters, many operations can be performed upon them, without thinking of the lines or the figure. And this circumstance proves of no small advantage: the mind is relieved, and spared as much as possible, that its whole force may be employed in overcoming the natural difficulty of the problem alone.
Upon the whole therefore the state of the comparison seems to be this; That the method of the ancients is fittest to begin our studies with, to form the mind and to establish proper habits; and that of the moderns to succeed, by extending our views beyond the present limits, and enabling us to make new discoveries and improvements.
Analysis is divided, with respect to its object, into that of finites, and that of infinites.
Analysis of finite quantities, is what is otherwise called algebra, or specious arithmetic.
Analysis of infinites, called also the new analysis, is that which is concerned in calculating the relations of quantities which are considered as infinite, or infinitely little; one of its chief branches being the method of fluxions, or the differential calculus. And the great advantage of the modern mathematicians over the ancients, arises chiefly from the use of this modern analysis.
Analysis of powers, is the same as resolving them into their roots, and is otherwise called evolution.
Analysis of curve lines, shews their constitution, nature and properties, their points of inflexion, station, retrogradation, variation, &c.