1701. The science of algebra, independently of any of its uses, has all the advantages which belong to mathematics in general as an object of study, and which it is not necessary to enumerate. Viewed either as a science of quantity, or as a language of symbols, it may be made of the greatest service to those who are sufficiently acquainted with arithmetic, and who have sufficient power of comprehension to enter fairly upon its difficulties.—De Morgan, A.

Elements of Algebra (London, 1837), Preface.

1702. Algebra is generous, she often gives more than is asked of her.—D’Alembert.

Quoted in Bulletin American Mathematical Society, Vol. 2 (1905), p. 285.

1703. The operations of symbolic arithmetick seem to me to afford men one of the clearest exercises of reason that I ever yet met with, nothing being there to be performed without strict and watchful ratiocination, and the whole method and progress of that appearing at once upon the paper, when the operation is finished, and affording the analyst a lasting, and, as it were, visible ratiocination.—Boyle, Robert.

Works (London, 1772), Vol. 3, p. 426.

1704. The human mind has never invented a labor-saving machine equal to algebra.—

The Nation, Vol. 33, p. 237.

1705. They that are ignorant of Algebra cannot imagine the wonders in this kind are to be done by it: and what further improvements and helps advantageous to other parts of knowledge the sagacious mind of man may yet find out, it is not easy to determine. This at least I believe, that the ideas of quantity are not those alone that are capable of demonstration and knowledge; and that other, and perhaps more useful, parts of contemplation, would afford us certainty, if vices, passions, and domineering interest did not oppose and menace such endeavours.—Locke, John.

An Essay concerning Human Understanding, Bk. 4, chap. 3, sect. 18.

1706. Algebra is but written geometry and geometry is but figured algebra.—Germain, Sophie.

MÉmoire sur la surfaces Élastiques.

1707. So long as algebra and geometry proceeded separately their progress was slow and their application limited, but when these two sciences were united, they mutually strengthened each other, and marched together at a rapid pace toward perfection.—Lagrange.

LeÇons ÉlÉmentaires sur les MathÉmatiques, LeÇon CinquiÈme.

1708. The laws of algebra, though suggested by arithmetic, do not depend on it. They depend entirely on the conventions by which it is stated that certain modes of grouping the symbols are to be considered as identical. This assigns certain properties to the marks which form the symbols of algebra. The laws regulating the manipulation of algebraic symbols are identical with those of arithmetic. It follows that no algebraic theorem can ever contradict any result which could be arrived at by arithmetic; for the reasoning in both cases merely applies the same general laws to different classes of things. If an algebraic theorem can be interpreted in arithmetic, the corresponding arithmetical theorem is therefore true.—Whitehead, A. N.

Universal Algebra (Cambridge, 1898), p. 2.

1709. That a formal science like algebra, the creation of our abstract thought, should thus, in a sense, dictate the laws of its own being, is very remarkable. It has required the experience of centuries for us to realize the full force of this appeal.—Mathews, G. B.

F. Spencer: Chapters on Aims and Practice of Teaching (London, 1899), p. 184.

1710. The rules of algebra may be investigated by its own principles, without any aid from geometry; and although in many cases the two sciences may serve to illustrate each other, there is not now the least necessity in the more elementary parts to call in the aid of the latter in expounding the former.—Chrystal, George.

Encyclopedia Britannica, 9th Edition; Article “Algebra”

1711. Algebra, as an art, can be of no use to any one in the business of life; certainly not as taught in the schools. I appeal to every man who has been through the school routine whether this be not the case. Taught as an art it is of little use in the higher mathematics, as those are made to feel who attempt to study the differential calculus without knowing more of the principles than is contained in books of rules.—De Morgan, A.

Elements of Algebra (London, 1837), Preface.

1712. We may always depend upon it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.—Clifford, W. K.

Common Sense in the Exact Sciences (London, 1885), chap. 1, sect. 7.

1713. The best review of arithmetic consists in the study of algebra.—Cajori, F.

Teaching and History of Mathematics in U. S. (Washington, 1896), p. 110.

1714. [Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions.... We will briefly say that Algebra is the Calculus of Functions, and Arithmetic the Calculus of Values.—Comte, A.

Philosophy of Mathematics [Gillespie] (New York, 1851), p. 55.

1715. ... the subject matter of algebraic science is the abstract notion of time; divested of, or not yet clothed with, any actual knowledge which we may possess of the real Events of History, or any conception which we may frame of Cause and Effect in Nature; but involving, what indeed it cannot be divested of, the thought of possible Succession, or of pure, ideal Progression.—Hamilton, W. R.

Graves’ Life of Hamilton (New York, 1882-1889), Vol. 3, p. 633.

1716. ... instead of seeking to attain consistency and uniformity of system, as some modern writers have attempted, by banishing this thought of time from the higher Algebra, I seek to attain the same object, by systematically introducing it into the lower or earlier parts of the science.—Hamilton, W. R.

Graves’ Life of Hamilton (New York, 1882-1889), Vol. 3, p. 634.

1717. The circumstances that algebra has its origin in arithmetic, however widely it may in the end differ from that science, led Sir Isaac Newton to designate it “Universal Arithmetic,” a designation which, vague as it is, indicates its character better than any other by which it has been attempted to express its functions—better certainly, to ordinary minds, than the designation which has been applied to it by Sir William Rowan Hamilton, one of the greatest mathematicians the world has seen since the days of Newton—“the Science of Pure Time;” or even than the title by which De Morgan would paraphrase Hamilton’s words—“the Calculus of Succession”—Chrystal, George.

Encyclopedia Britannica, 9th Edition; Article “Algebra”

1718. Time is said to have only one dimension, and space to have three dimensions.... The mathematical quaternion partakes of both these elements; in technical language it may be said to be “time plus space,” or “space plus time:” and in this sense it has, or at least involves a reference to, four dimensions....

—Hamilton, W. R.

Graves’ Life of Hamilton (New York, 1882-1889), Vol. 3, p. 635.

1719. It is confidently predicted, by those best qualified to judge, that in the coming centuries Hamilton’s Quaternions will stand out as the great discovery of our nineteenth century. Yet how silently has the book taken its place upon the shelves of the mathematician’s library! Perhaps not fifty men on this side of the Atlantic have seen it, certainly not five have read it.—Hill, Thomas.

North American Review, Vol. 85, p. 223.

1720. I think the time may come when double algebra will be the beginner’s tool; and quaternions will be where double algebra is now. The Lord only knows what will come above the quaternions.—De Morgan, A.

Graves’ Life of Hamilton (New York, 1882-1889), Vol. 3, p. 493.

1721. Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.—Thomson, William.

Thompson, S. P.: Life of Lord Kelvin (London, 1910), p. 1138.

1722. The whole affair [quaternions] has in respect to mathematics a value not inferior to that of “Volapuk” in respect to language.—Thomson, William.

Thompson, S. P.: Life of Lord Kelvin (London, 1910), p. 1138.

1723. A quaternion of maladies! Do send me some formula by help of which I may so doctor them that they may all become imaginary or positively equal to nothing.—Sedgwick.

Graves’ Life of Hamilton (New York, 1882-1889), Vol. 3, p. 2.

1724. If nothing more could be said of Quaternions than that they enable us to exhibit in a singularly compact and elegant form, whose meaning is obvious at a glance on account of the utter inartificiality of the method, results which in the ordinary Cartesian co-ordinates are of the utmost complexity, a very powerful argument for their use would be furnished. But it would be unjust to Quaternions to be content with such a statement; for we are fully entitled to say that in all cases, even in those to which the Cartesian methods seem specially adapted, they give as simple an expression as any other method; while in the great majority of cases they give a vastly simpler one. In the common methods a judicious choice of co-ordinates is often of immense importance in simplifying an investigation; in Quaternions there is usually no choice, for (except when they degrade to mere scalars) they are in general utterly independent of any particular directions in space, and select of themselves the most natural reference lines for each particular problem.—Tait, P. G.

Presidential Address British Association for the Advancement of Science (1871); Nature, Vol. 4, p. 270.

1725. Comparing a Quaternion investigation, no matter in what department, with the equivalent Cartesian one, even when the latter has availed itself to the utmost of the improvements suggested by Higher Algebra, one can hardly help making the remark that they contrast even more strongly than the decimal notation with the binary scale, or with the old Greek arithmetic—or than the well-ordered subdivisions of the metrical system with the preposterous no-systems of Great Britain, a mere fragment of which (in the form of Table of Weights and Measures) form, perhaps the most effective, if not the most ingenious, of the many instruments of torture employed in our elementary teaching.—Tait, P. G.

Presidential Address British Association for the Advancement of Science (1871); Nature, Vol. 4, p. 271.

1726. It is true that, in the eyes of the pure mathematician, Quaternions have one grand and fatal defect. They cannot be applied to space of n dimensions, they are contented to deal with those poor three dimensions in which mere mortals are doomed to dwell, but which cannot bound the limitless aspirations of a Cayley or a Sylvester. From the physical point of view this, instead of a defect, is to be regarded as the greatest possible recommendation. It shows, in fact, Quaternions to be the special instrument so constructed for application to the Actual as to have thrown overboard everything which is not absolutely necessary, without the slightest consideration whether or no it was thereby being rendered useless for application to the Inconceivable.—Tait, P. G.

Presidential Address British Association for the Advancement of Science (1871); Nature, Vol. 4, p. 271.

1727. There is an old epigram which assigns the empire of the sea to the English, of the land to the French, and of the clouds to the Germans. Surely it was from the clouds that the Germans fetched + and -; the ideas which these symbols have generated are much too important to the welfare of humanity to have come from the sea or from the land.—Whitehead, A. N.

An Introduction to Mathematics (New York, 1911), p. 86.

1728. Now as to what pertains to these Surd numbers (which, as it were by way of reproach and calumny, having no merit of their own are also styled Irrational, Irregular, and Inexplicable) they are by many denied to be numbers properly speaking, and are wont to be banished from arithmetic to another Science, (which yet is no science) viz. algebra.—Barrow, Isaac.

Mathematical Lectures (London, 1734), p. 44.

1729. If it is true as Whewell says, that the essence of the triumphs of science and its progress consists in that it enables us to consider evident and necessary, views which our ancestors held to be unintelligible and were unable to comprehend, then the extension of the number concept to include the irrational, and we will at once add, the imaginary, is the greatest forward step which pure mathematics has ever taken.—Hankel, Hermann.

Theorie der Complexen Zahlen (Leipzig, 1867), p. 60.

1730. That this subject [of imaginary magnitudes] has hitherto been considered from the wrong point of view and surrounded by a mysterious obscurity, is to be attributed largely to an ill-adapted notation. If for instance, +1,-1, v-1 had been called direct, inverse, and lateral units, instead of positive, negative, and imaginary (or even impossible) such an obscurity would have been out of question.—Gauss, C. F.

Theoria residiorum biquadraticorum, Commentatio secunda; Werke, Bd. 2 (Goettingen, 1863), p. 177.

1731. ... the imaginary, this bosom-child of complex mysticism.—DÜhring, Eugen.

Kritische Geschichte der allgemeinen Principien der Mechanik (Leipzig, 1877), p. 517.

1732. Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers).—Fine, H. B.

The Number-System of Algebra (Boston, 1890), p. 36.

1733. This symbol [v-1] is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power. The immortal author of quaternions has shown that there are other significations which may attach to the symbol in other cases. But the strongest use of the symbol is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry.—Peirce, Benjamin.

On the Uses and Transformations of Linear Algebras; American Journal of Mathematics, Vol. 4 (1881), p. 216.

1734. The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the masterkeys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments.—Hill, Thomas.

North American Review, Vol. 85, p. 235.

1735. All the fruitful uses of imaginaries, in Geometry, are those which begin and end with real quantities, and use imaginaries only for the intermediate steps. Now in all such cases, we have a real spatial interpretation at the beginning and end of our argument, where alone the spatial interpretation is important; in the intermediate links, we are dealing in purely algebraic manner with purely algebraic quantities, and may perform any operations which are algebraically permissible. If the quantities with which we end are capable of spatial interpretation, then, and only then, our results may be regarded as geometrical. To use geometrical language, in any other case, is only a convenient help to the imagination. To speak, for example, of projective properties which refer to the circular points, is a mere memoria technica for purely algebraical properties; the circular points are not to be found in space, but only in the auxiliary quantities by which geometrical equations are transformed. That no contradictions arise from the geometrical interpretation of imaginaries is not wonderful; for they are interpreted solely by the rules of Algebra, which we may admit as valid in their interpretation to imaginaries. The perception of space being wholly absent, Algebra rules supreme, and no inconsistency can arise.—Russell, Bertrand.

Foundations of Geometry (Cambridge, 1897), p. 45.

1736. Indeed, if one understands by algebra the application of arithmetic operations to composite magnitudes of all kinds, whether they be rational or irrational number or space magnitudes, then the learned Brahmins of Hindostan are the true inventors of algebra.—Hankel, Hermann.

Geschichte der Mathematik im Altertum und Mittelalter (Leipzig, 1874), p. 195.

1737. It is remarkable to what extent Indian mathematics enters into the science of our time. Both the form and the spirit of the arithmetic and algebra of modern times are essentially Indian and not Grecian.—Cajori, F.

History of Mathematics (New York, 1897), p. 100.

1738. There are many questions in this science [algebra] which learned men have to this time in vain attempted to solve; and they have stated some of these questions in their writings, to prove that this science contains difficulties, to silence those who pretend they find nothing in it above their ability, to warn mathematicians against undertaking to answer every question that may be proposed, and to excite men of genius to attempt their solution. Of these I have selected seven.

1. To divide 10 into two parts, such, that when each part is added to its square-root and the sums multiplied together, the product is equal to the supposed number.

2. What square is that, which being increased or diminished by 10, the sum and remainder are both square numbers?

3. A person said he owed to Zaid 10 all but the square-root of what he owed to Amir, and that he owed Amir 5 all but the square-root of what he owed Zaid.

4. To divide a cube number into two cube numbers.

5. To divide 10 into two parts such, that if each is divided by the other, and the two quotients are added together, the sum is equal to one of the parts.

6. There are three square numbers in continued geometric proportion, such, that the sum of the three is a square number.

7. There is a square, such, that when it is increased and diminished by its root and 2, the sum and the difference are squares.—Khulasat-al-Hisab.

Algebra; quoted in Hutton: A Philosophical and Mathematical Dictionary (London, 1815), Vol. 1, p. 70.

1739. The solution of such questions as these [referring to the solution of cubic equations] depends on correct judgment, aided by the assistance of God.—Bija Ganita.

Quoted in Hutton: A Philosophical and Mathematical Dictionary (London, 1815), Vol. 1, p. 65.

1740. For what is the theory of determinants? It is an algebra upon algebra; a calculus which enables us to combine and foretell the results of algebraical operations, in the same way as algebra itself enables us to dispense with the performance of the special operations of arithmetic. All analysis must ultimately clothe itself under this form.—Sylvester, J. J.

Philosophical Magazine, Vol. 1, (1851), p. 300; Collected Mathematical Papers, Vol. 1, p. 247.

1741.

Fuchs.

Fast mÖcht’ ich nun moderne Algebra studieren.

Meph.

Ich wÜnschte nicht euch irre zu fÜhren.

Was diese Wissenschaft betrifft,

Es ist so schwer, die leere Form zu meiden,

Und wenn ihr es nicht recht begrifft,

VermÖgt die Indices ihr kaum zu unterscheiden.

Am Besten ist’s, wenn ihr nur Einem traut

Und auf des Meister’s Formeln baut.

Im Ganzen—haltet euch an die Symbole.

Dann geht ihr zu der Forschung Wohle

Ins sichre Reich der Formeln ein.

Fuchs.

Ein Resultat muss beim Symbole sein?

Meph.

Schon gut! Nur muss man sich nicht alzu Ängstlich quÄlen.

Denn eben, wo die Resultate fehlen,

Stellt ein Symbol zur rechten Zeit sich ein.

Symbolisch lÄsst sich alles schreiben,

MÜsst nur im Allgemeinen bleiben.