His vention Parliament, and was a second time returned in 1701. theological writings evince his sincere belief in the Records of the Christian Revelation, and a passage in the general scholium at the 1 The following is a translation of the passage: "This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One, especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all other systems. And lest the systems of the fixed stars should, by their gravity, fall on each other mutually, He has placed these systems at immense distances one from another. 66 This Being governs all things, not as the soul of the world, but as Lord over all. And on account of his dominion He is wont to be called Lord God, яаνтокράTwp, or Universal Ruler. For God is a relative word, and has a respect to servants; and Deity is the dominion of God, not over His own body, as those imagine who fancy God to be the soul of the world, but over servants. The supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of gods, and Lord of lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of gods; we do not say, my Infinite, or my Perfect : these are titles which have no respect to servants. The word God usually signifies Lord; but every lord is not a god. It is the dominion of a spiritual being which constitutes a god; a true, supreme, or imaginary dominion makes a true, supreme, or imaginary god. And from His true dominion it follows that the true God is a Living, Intelligent, and Powerful Being; and from His other perfections, that He is supreme or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, His duration reaches from eternity to eternity; His presence from infinity to infinity; He governs all things, and knows all things that are or can be done. He is not eternity nor infinity, but eternal and infinite; He is not duration or space, but He endures and is present. He endures for ever, and is everywhere present; and by existing always and everywhere, He constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is everywhere, certainly the Maker and Lord of all things cannot be never and nowhere. Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent parts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and everywhere. He is omnipresent, not virtually only, but also substantially; for virtue cannot subsist without substance. In Him are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. "It is allowed by all that the supreme God exists necessarily; and by the same necessity He exists always and everywhere. Whence also He is all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly unknown to us. As a blind man has no idea of colours, so have we no idea of the manner by which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither be seen, nor end of the second edition of his great work, exhibits his conception of the Deity as the Creator and Ruler of the universe. In the year 1715, Dr. Brook Taylor, of St. John's College, Cambridge, published his "Methodus Incrementorum Directa et Inversa” in a quarto volume. He gives to the differences of two variable quantities the name of increments or decrements, whether considered finite or infinitely small, of two consecutive terms in a series formed according to a given law. When these differences are indefinitely small, the calculus of them, both direct and inverse, belongs to the Differential Calculus. But when the differences are finite, the method of finding the relations they bear to the quantities that produce them, forms a new kind of calculus, the first principles of which were given by Dr. Brook Taylor himself. His work is extremely concise and somewhat obscure. M. Nicole, a French mathematician, has clearly unfolded the method for resolving finite differences, which may be read in the Memoirs of the French Academy of Sciences for 1717 and 1728; and these articles may be considered as the first elementary treatise on the Integral Calculus with finite differences. In one of the corollaries to the seventh proposition in his work, Dr. Brook Taylor has given a new theorem of the highest importance and of very extensive use in mathematical reasonings. It consists of a series in ascending powers of the increment, to express what any function of a variable quantity becomes, when the variable quantity receives any increment whatever. This famous theorem is called "Taylor's Theorem," from the name of its inventor. It was originally expressed in the fluxional notation; but in later times it has been expressed in the notation of the Differential Calculus, and extended to functions of two or more variable quantities. heard, nor touched; nor ought He to be worshipped under the representation of any corporeal thing. We have ideas of His attributes, but what the real substance of anything is, we know not. In bodies we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known, either by our senses, or by any reflex act of our minds; much less then have we any idea of the substance of God. We know Him only by His most wise and excellent contrivances of things, and final causes; we admire Him for His perfections, but we reverence and adore Him on account of His dominion. For we adore Him as His servants, and a God without dominion, providence, and final causes, is nothing else but fate and nature. Blind metaphysical necessity, which is certainly the same always and everywhere, could produce no variety of things. All that diversity of natural things which we find, suited to different times and places, could arise from nothing but the ideas and will of a Being necessarily existing. But by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build. For all our notions of God are taken from the ways of mankind, by a certain similitude which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to natural philosophy." In the first chapters of his "Institutions," Euler gave a lucid explanation of the principles of this subject; it was also treated by Emerson in his Method of Increments, which was published in 1763. The formula which is generally known as Maclaurin's Theorem, will be found in p. 610 of the second volume of his treatise on Fluxions, published in 1742. It may, however, be noted that the same formula appears in p. 102 of Stirling's" Methodus Differentialis," which was published in 1730. This theorem may easily be deduced from Taylor's Theorem (of which it is a particular case), and is useful in finding the expansion of any function of a variable in a series of ascending powers of that variable. The labours of the Bernoullis in connection with Leibnitz contributed very largely to the progress of the new calculus on the Continent. Soon after the early essays of Leibnitz appeared in the Leipsic Acts, in which the art and method of his calculus were not fully disclosed, James Bernoulli and his brother John, having read these papers, saw the advantages of his method, and were induced to attempt to search out the full meaning of Leibnitz. Their efforts ended in such success, that Leibnitz himself is reported to have declared that the invention belonged to them as much as to himself. The contributions of John Bernoulli are printed in the Leipsic Acts and in the Memoirs of the French Academy. His collected works were published at Geneva in four quarto volumes in 1742. The valuable papers of James Bernoulli on the calculus and various other subjects were printed in the Leipsic Acts. His works, that had been previously printed, were collected and published at Geneva in two quarto volumes, in 1744. His papers on the Integral Calculus give an excellent exposition of the subject, especially in the formation and integration of differential equations. Daniel Bernoulli, one of the sons of John Bernoulli, was an eminent mathematician, and during his lifetime he was successful in bearing off ten prizes from the Academy of Sciences, which were contended for by the greatest mathematicians of Europe. In the year 1734 he divided the honour of one of them with his father, which caused an estrangement between them. John de Rond D'Alembert was one of the most eminent scholars and mathematicians of his age. He was born in 1717, and died in 1783. His literary and scientific labours embraced numerous subjects on philology, history, and philosophy. His new principle of the calculus of differences he exemplified in the discourse on the general theory of the winds, for which the prize medal of the Academy of Berlin was adjudged in 1746. He also applied the same principle to the problems of vibrating chords and the propagation of sound. His writings contain some valuable papers on the Integral Calculus. The mathematical writings of D'Alembert were published under the title of "Opuscules Mathematiques, ou Memoires sur differens sujets de Géométrie, de Méchaniques, d'Optiques, d'Astronomie," in eight quarto volumes, between the years 1761 and 1780. Euler published his "Institutiones Calculi Differentialis" in 1755, and his "Institutiones Calculi Integralis" in 1768-70. In the preface to the former he describes the Differential Calculus as "the art of finding the vanishing increments that any functions acquire, when we attribute to the variable quantity of which they are functions a vanishing increment." His "Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes" was published at an earlier period, in the year 1744. It has been seen that there is a marked difference between the original conceptions of Newton and Leibnitz with respect to the foundations of the calculus. The differentials of Leibnitz were indefinitely small quantities added to, or subtracted from the variable quantity. Newton introduced into his method of fluxions the conception of time and motion-subjects foreign to geometry, and belonging to the science of dynamics. Thus he employed motion as the means of connecting its principles with ordinary Algebra; whereas Leibnitz considered the increase or decrease of a variable to arise from the continued addition or subtraction of independently small parts without any reference to time. Mathematicians have differed in opinion as to the most scientific method of laying down the principles of the calculus, and different attempts have been made to substitute other methods than those of Newton and Leibnitz, but none of them has been more favourably considered than that of La Grange. La Grange proposed, in the Memoirs of the Berlin Academy for 1772, to make the calculus dependent on the ordinary principles of Algebra, divesting the subject of the consideration of the limiting ratio of indefinitely small increments. He subsequently extended his views of the subject, and published in 1797 his method in a volume entitled "Théorie des Fonctions Analytiques, contenant les principes du Calcul Différentiel, dégagés de tout consideration d'infiniment petits, d'évanouissants; de limits et de fluxions et reduits a l'analyse Algebrique des quantitiés finies." He further elucidated and completed his views in another work which he published in 1801, with the title, "Leçons sur le Calcul des Fonctions." He makes the basis of his method Taylor's Theorem, the form of which he obtains by the processes of ordinary Algebra. As in the method of finite differences, he considers differentials as finite quantities, always indeterminate, so that they can be made as small as we please. He has also given theorems for the determination of the limits between which lies the remainder of Taylor's series after a finite number of terms has been found. La Grange devised a formula that bears his name, which is useful for the development of implicit functions. It first appeared in the Berlin Memoirs for the year 1770. A more general formula for the developement of implicit functions was discovered by La Place, and was first printed in the Memoirs of the French Academy of Sciences for 1777. La Grange's theorem is only a particular case of the theorem of La Place. The calculus of variations, an important improvement in the calculus, is due to La Grange. His papers on the subject are printed in the second and fourth volumes of the Miscel. Taur. 1760 to 1769. The principal object of this calculus is to solve generally certain questions of maxima and minima. In the ordinary questions of the subject, it is required to find the values which must be assigned to different variables which enter into a proposed finite function of these variables, so that this function may attain its greatest or least possible value. In the calculus of variations, on the contrary, it is required to find the ratios between the variables, in order to satisfy the condition of a maximum or minimum. Besides, the function which must be a maximum or minimum is not, as in ordinary questions, solely composed of finite quantities, but it must be the integral simply marked of a differential function which is not to be integrated. In 1797, M. Carnot published his "Reflexions on the Metaphysical Principles of the Infinitesimal Analysis." It was translated into English by the Rev. W. R. Browell, M.A., Fellow of Pembroke College, Oxford, and published at Oxford in 1832. Every expression of Algebra must necessarily give an idea rather of a set of operations to be performed upon quantities than of the quantity itself which results from performing these operations. Nothing is more difficult than the necessity of considering a complicated literal expression as the value of an unknown quantity instead of the exhibition of the processes by which the value is to be obtained, so soon as the subjects of operation shall have received. specific values.1 Many of the calculations with which we are familiar consist of two parts, a direct and an inverse; as when we consider an exponent of a quantity. For instance, to raise any number to a given power is the direct operation; to extract a given root of any number is the inverse method. The Differential Calculus, which is a direct method, naturally gave rise to the Integral, which is its inverse; the same remark is applicable to finite differences. In all these cases the inverse method is by far the most difficult, and it might perhaps be added the most useful. If an unknown quantity be given by means of an equation, it becomes a question to determine its value; similarly, if an unknown function be given by means of any functional equation, it is required to assign its form. In the first case, it is quantity which is to be determined; in the second, it is the form assumed by quantity, that 1 Encyclop. Metropol., ii.; Babbage, p. 366. |