Robert Edouard Moritz

1801. The science of figures is most glorious and beautiful. But how inaptly it has received the name geometry!—Frischlinus, N.

Dialog 1.

1802. Plato said that God geometrizes continually.—Plutarch.

Convivialium disputationum, liber 8, 2.

1803. ?de?? ??e??t??t?? e?s?t? ?? t?? st????. [Let no one ignorant of geometry enter my door.]—Plato.

Tzetzes, Chiliad, 8, 972.

1804. All the authorities agree that he [Plato] made a study of geometry or some exact science an indispensable preliminary to that of philosophy. The inscription over the entrance to his school ran “Let none ignorant of geometry enter my door,” and on one occasion an applicant who knew no geometry is said to have been refused admission as a student.—Ball, W. W. R.

History of Mathematics (London, 1901), p. 45.

1805. Form and size constitute the foundation of all search for truth.—Parker, F. W.

Talks on Pedagogics (New York, 1894), p. 72.

1806. At present the science [of geometry] is in flat contradiction to the language which geometricians use, as will hardly be denied by those who have any acquaintance with the study: for they speak of finding the side of a square, and applying and adding, and so on, as if they were engaged in some business, and as if all their propositions had a practical end in view: whereas in reality the science is pursued wholly for the sake of knowledge.

Certainly, he said.

Then must not a further admission be made?

What admission?

The admission that this knowledge at which geometry aims is of the eternal, and not of the perishing and transient.

That may be easily allowed. Geometry, no doubt, is the knowledge of what eternally exists.

Then, my noble friend, geometry will draw the soul towards truth, and create the mind of philosophy, and raise up that which is now unhappily allowed to fall down.—Plato.

Republic [Jowett-Davies], Bk. 7, p. 527.

1807. Among them [the Greeks] geometry was held in highest honor: nothing was more glorious than mathematics. But we have limited the usefulness of this art to measuring and calculating.—Cicero.

Tusculanae Disputationes, 1, 2, 5.

1808.

Geometria,

Through which a man hath the sleight

Of length, and brede, of depth, of height.

—Gower, John.

Confessio Amantis, Bk. 7.

1809. Geometrical truths are in a way asymptotes to physical truths, that is to say, the latter approach the former indefinitely near without ever reaching them exactly.—D’Alembert.

Quoted in RebiÈre: MathÉmatiques et MathÉmaticiens (Paris, 1898), p. 10.

1810. Geometry exhibits the most perfect example of logical stratagem.—Buckle, H. T.

History of Civilization in England (New York, 1891), Vol. 2, p. 342.

1811. It is the glory of geometry that from so few principles, fetched from without, it is able to accomplish so much.—Newton.

Philosophiae Naturalis Principia Mathematica, Praefat.

1812. Geometry is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. But the rigor of this science is carried one step further; for no property, however evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to demonstrate all geometrical truths with the smallest possible number of assumptions.—De Morgan, A.

On the Study and Difficulties of Mathematics (Chicago, 1902), p. 231.

1813. Geometry is a true natural science:—only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Every body studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone.—Comte, A.

Positive Philosophy [Martineau], Bk. 1, chap. 3.

1814. Geometry in every proposition speaks a language which experience never dares to utter; and indeed of which she but half comprehends the meaning. Experience sees that the assertions are true, but she sees not how profound and absolute is their truth. She unhesitatingly assents to the laws which geometry delivers, but she does not pretend to see the origin of their obligation. She is always ready to acknowledge the sway of pure scientific principles as a matter of fact, but she does not dream of offering her opinion on their authority as a matter of right; still less can she justly claim to herself the source of that authority.—Whewell, William.

The Philosophy of the Inductive Sciences, Part 1, Bk. 1, chap. 6, sect. 1 (London, 1858).

1815. Geometry is the science created to give understanding and mastery of the external relations of things; to make easy the explanation and description of such relations and the transmission of this mastery.—Halsted, G. B.

Proceedings of the American Association for the Advancement of Science (1904), p. 359.

1816. A mathematical point is the most indivisible and unique thing which art can present.—Donne, John.

Letters, 21.

1817. It is certain that from its completeness, uniformity and faultlessness, from its arrangement and progressive character, and from the universal adoption of the completest and best line of argument, Euclid’s “Elements” stand pre-eminently at the head of all human productions. In no science, in no department of knowledge, has anything appeared like this work: for upward of 2000 years it has commanded the admiration of mankind, and that period has suggested little toward its improvement.—Kelland, P.

Lectures on the Principles of Demonstrative Mathematics (London, 1843), p. 17.

1818. In comparing the performance in Euclid with that in Arithmetic and Algebra there could be no doubt that Euclid had made the deepest and most beneficial impression: in fact it might be asserted that this constituted by far the most valuable part of the whole training to which such persons [students, the majority of which were not distinguished for mathematical taste and power] were subjected.—Todhunter, I.

Essay on Elementary Geometry; Conflict of Studies and other Essays (London, 1873), p. 167.

1819. In England the geometry studied is that of Euclid, and I hope it never will be any other; for this reason, that so much has been written on Euclid, and all the difficulties of geometry have so uniformly been considered with reference to the form in which they appear in Euclid, that the study of that author is a better key to a great quantity of useful reading than any other.—De Morgan, A.

Elements of Algebra (London, 1837), Introduction.

1820. This book [Euclid] has been for nearly twenty-two centuries the encouragement and guide of that scientific thought which is one thing with the progress of man from a worse to a better state. The encouragement; for it contained a body of knowledge that was really known and could be relied on, and that moreover was growing in extent and application. For even at the time this book was written—shortly after the foundation of the Alexandrian Museum—Mathematics was no longer the merely ideal science of the Platonic school, but had started on her career of conquest over the whole world of Phenomena. The guide; for the aim of every scientific student of every subject was to bring his knowledge of that subject into a form as perfect as that which geometry had attained. Far up on the great mountain of Truth, which all the sciences hope to scale, the foremost of that sacred sisterhood was seen, beckoning for the rest to follow her. And hence she was called, in the dialect of the Phythagoreans, “the purifier of the reasonable soul”—Clifford, W. K.

Lectures and Essays (London, 1901), Vol. 1, p. 354.

1821. [Euclid] at once the inspiration and aspiration of scientific thought.—Clifford, W. K.

Lectures and Essays (London, 1901), Vol 1, p. 355.

1822. The “elements” of the Great Alexandrian remain for all time the first, and one may venture to assert, the only perfect model of logical exactness of principles, and of rigorous development of theorems. If one would see how a science can be constructed and developed to its minutest details from a very small number of intuitively perceived axioms, postulates, and plain definitions, by means of rigorous, one would almost say chaste, syllogism, which nowhere makes use of surreptitious or foreign aids, if one would see how a science may thus be constructed one must turn to the elements of Euclid.—Hankel, H.

Die Entwickelung der Mathematik in den letzten Jahrhunderten (TÜbingen, 1884), p. 7.

1823. If we consider him [Euclid] as meaning to be what his commentators have taken him to be, a model of the most unscrupulous formal rigour, we can deny that he has altogether succeeded, though we admit that he made the nearest approach.—De Morgan, A.

Smith’s Dictionary of Greek and Roman Biography and Mythology (London, 1902); Article “Eucleides”

1824. The Elements of Euclid is as small a part of mathematics as the Iliad is of literature; or as the sculpture of Phidias is of the world’s total art.—Keyser, C. J.

Lectures on Science, Philosophy and Art (New York, 1908), p. 8.

1825. I should rejoice to see ... Euclid honourably shelved or buried “deeper than did ever plummet sound” out of the schoolboys’ reach; morphology introduced into the elements of algebra; projection, correlation, and motion accepted as aids to geometry; the mind of the student quickened and elevated and his faith awakened by early initiation into the ruling ideas of polarity, continuity, infinity, and familiarization with the doctrines of the imaginary and inconceivable.—Sylvester, J. J.

A Plea for the Mathematician; Nature, Vol. 1, p. 261.

1826. The early study of Euclid made me a hater of geometry, ... and yet, in spite of this repugnance, which had become a second nature in me, whenever I went far enough into any mathematical question, I found I touched, at last, a geometrical bottom.—Sylvester, J. J.

A Plea for the Mathematician; Nature, Vol. 1, p. 262.

1827. Newton had so remarkable a talent for mathematics that Euclid’s Geometry seemed to him “a trifling book,” and he wondered that any man should have taken the trouble to demonstrate propositions, the truth of which was so obvious to him at the first glance. But, on attempting to read the more abstruse geometry of Descartes, without having mastered the elements of the science, he was baffled, and was glad to come back again to his Euclid.—Parton, James.

Sir Isaac Newton.

1828. As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into the vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous.—Heaviside, Oliver.

Electro-Magnetic Theory (London, 1893), Vol. 1, p. 148.

1829. Geometry is nothing if it be not rigorous, and the whole educational value of the study is lost, if strictness of demonstration be trifled with. The methods of Euclid are, by almost universal consent, unexceptionable in point of rigour.—Smith, H. J. S.

Nature, Vol. 8, p. 450.

1830. To seek for proof of geometrical propositions by an appeal to observation proves nothing in reality, except that the person who has recourse to such grounds has no due apprehension of the nature of geometrical demonstration. We have heard of persons who convince themselves by measurement that the geometrical rule respecting the squares on the sides of a right-angles triangle was true: but these were persons whose minds had been engrossed by practical habits, and in whom speculative development of the idea of space had been stifled by other employments.—Whewell, William.

The Philosophy of the Inductive Sciences, (London, 1858), Part 1, Bk. 2, chap. 1, sect. 4.

1831. No one has ever given so easy and natural a chain of geometrical consequences [as Euclid]. There is a never-erring truth in the results.—De Morgan, A.

Smith’s Dictionary of Greek and Roman Biography and Mythology (London, 1902); Article “Eucleides”

1832. Beyond question, Egyptian geometry, such as it was, was eagerly studied by the early Greek philosophers, and was the germ from which in their hands grew that magnificent science to which every Englishman is indebted for his first lessons in right seeing and thinking.—Gow, James.

A Short History of Greek Mathematics (Cambridge, 1884), p. 131.

1833.

A figure and a step onward:

Not a figure and a florin.

—Motto of the Pythagorean Brotherhood.

W. B. Frankland: Story of Euclid (London, 1902), p. 33.

1834. The doctrine of proportion, as laid down in the fifth book of Euclid, is, probably, still unsurpassed as a masterpiece of exact reasoning; although the cumbrousness of the forms of expression which were adopted in the old geometry has led to the total exclusion of this part of the elements from the ordinary course of geometrical education. A zealous defender of Euclid might add with truth that the gap thus created in the elementary teaching of mathematics has never been adequately supplied.—Smith, H. J. S.

Presidential Address British Association for the Advancement of Science (1873); Nature, Vol. 8, p. 451.

1835. The Definition in the Elements, according to Clavius, is this: Magnitudes are said to be in the same Reason [ratio], a first to a second, and a third to a fourth, when the Equimultiples of the first and third according to any Multiplication whatsoever are both together either short of, equal to, or exceed the Equimultiples of the second and fourth, if those be taken, which answer one another.... Such is Euclid’s Definition of Proportions; that scare-Crow at which the over modest or slothful Dispositions of Men are generally affrighted: they are modest, who distrust their own Ability, as soon as a Difficulty appears, but they are slothful that will not give some Attention for the learning of Sciences; as if while we are involved in Obscurity we could clear ourselves without Labour. Both of which Sorts of Persons are to be admonished, that the former be not discouraged, nor the latter refuse a little Care and Diligence when a Thing requires some Study.—Barrow, Isaac.

Mathematical Lectures (London, 1734), p. 388.

1836. Of all branches of human knowledge, there is none which, like it [geometry] has sprung a completely armed Minerva from the head of Jupiter; none before whose death-dealing Aegis doubt and inconsistency have so little dared to raise their eyes. It escapes the tedious and troublesome task of collecting experimental facts, which is the province of the natural sciences in the strict sense of the word: the sole form of its scientific method is deduction. Conclusion is deduced from conclusion, and yet no one of common sense doubts but that these geometrical principles must find their practical application in the real world about us. Land surveying, as well as architecture, the construction of machinery no less than mathematical physics, are continually calculating relations of space of the most varied kinds by geometrical principles; they expect that the success of their constructions and experiments shall agree with their calculations; and no case is known in which this expectation has been falsified, provided the calculations were made correctly and with sufficient data.—Helmholtz, H.

The Origin and Significance of Geometrical Axioms; Popular Scientific Lectures [Atkinson], Second Series (New York, 1881), p. 27.

1837. The amazing triumphs of this branch of mathematics [geometry] show how powerful a weapon that form of deduction is which proceeds by an artificial reparation of facts, in themselves inseparable.—Buckle, H. T.

History of Civilization in England (New York, 1891), Vol. 2, p. 343.

1838. Every theorem in geometry is a law of external nature, and might have been ascertained by generalizing from observation and experiment, which in this case resolve themselves into comparisons and measurements. But it was found practicable, and being practicable was desirable, to deduce these truths by ratiocination from a small number of general laws of nature, the certainty and universality of which was obvious to the most careless observer, and which compose the first principles and ultimate premises of the science.—Mill, J. S.

System of Logic, Bk. 3, chap. 24, sect. 7.

1839. All such reasonings [natural philosophy, chemistry, agriculture, political economy, etc.] are, in comparison with mathematics, very complex; requiring so much more than that does, beyond the process of merely deducing the conclusion logically from the premises: so that it is no wonder that the longest mathematical demonstration should be much more easily constructed and understood, than a much shorter train of just reasoning concerning real facts. The former has been aptly compared to a long and steep, but even and regular, flight of steps, which tries the breath, and the strength, and the perseverance only; while the latter resembles a short, but rugged and uneven, ascent up a precipice, which requires a quick eye, agile limbs, and a firm step; and in which we have to tread now on this side, now on that—ever considering as we proceed, whether this or that projection will afford room for our foot, or whether some loose stone may not slide from under us. There are probably as many steps of pure reasoning in one of the longer of Euclid’s demonstrations, as in the whole of an argumentative treatise on some other subject, occupying perhaps a considerable volume.—Whately, R.

Elements of Logic, Bk. 4, chap. 2, sect. 5.

1840.

[Geometry] that held acquaintance with the stars,

And wedded soul to soul in purest bond

Of reason, undisturbed by space or time.

—Wordsworth.

The Prelude, Bk. 5.

1841. The statement that a given individual has received a sound geometrical training implies that he has segregated from the whole of his sense impressions a certain set of these impressions, that he has eliminated from their consideration all irrelevant impressions (in other words, acquired a subjective command of these impressions), that he has developed on the basis of these impressions an ordered and continuous system of logical deduction, and finally that he is capable of expressing the nature of these impressions and his deductions therefrom in terms simple and free from ambiguity. Now the slightest consideration will convince any one not already conversant with the idea, that the same sequence of mental processes underlies the whole career of any individual in any walk of life if only he is not concerned entirely with manual labor; consequently a full training in the performance of such sequences must be regarded as forming an essential part of any education worthy of the name. Moreover the full appreciation of such processes has a higher value than is contained in the mental training involved, great though this be, for it induces an appreciation of intellectual unity and beauty which plays for the mind that part which the appreciation of schemes of shape and color plays for the artistic faculties; or, again, that part which the appreciation of a body of religious doctrine plays for the ethical aspirations. Now geometry is not the sole possible basis for inculcating this appreciation. Logic is an alternative for adults, provided that the individual is possessed of sufficient wide, though rough, experience on which to base his reasoning. Geometry is, however, highly desirable in that the objective bases are so simple and precise that they can be grasped at an early age, that the amount of training for the imagination is very large, that the deductive processes are not beyond the scope of ordinary boys, and finally that it affords a better basis for exercise in the art of simple and exact expression than any other possible subject of a school course.—Carson, G. W. L.

The Functions of Geometry as a Subject of Education (Tonbridge, 1910), p. 3.

1842. It seems to me that the thing that is wanting in the education of women is not the acquaintance with any facts, but accurate and scientific habits of thought, and the courage to think that true which appears unlikely. And for supplying this want there is a special advantage in geometry, namely that it does not require study of a physically laborious kind, but rather that rapid intuition which women certainly possess; so that it is fit to become a scientific pursuit for them.—Clifford, W. K.

Quoted by Pollock in Clifford’s Lectures and Essays (London, 1901), Vol. 1, Introduction, p. 43.

1843.

On the lecture slate

The circle rounded under female hands

With flawless demonstration.

—Tennyson.

The Princess, II, l. 493.

1844. It is plain that that part of geometry which bears upon strategy does concern us. For in pitching camps, or in occupying positions, or in closing or extending the lines of an army, and in all the other manoeuvres of an army whether in battle or on the march, it will make a great difference to a general, whether he is a geometrician or not.—Plato.

Republic, Bk. 7, p. 526.

1845. Then nothing should be more effectually enacted, than that the inhabitants of your fair city should learn geometry. Moreover the science has indirect effects, which are not small.

Of what kind are they? he said.

There are the military advantages of which you spoke, I said; and in all departments of study, as experience proves, any one who has studied geometry is infinitely quicker of apprehension.—Plato.

Republic [Jowett], Bk. 7, p. 527.

1846. It is doubtful if we have any other subject that does so much to bring to the front the danger of carelessness, of slovenly reasoning, of inaccuracy, and of forgetfulness as this science of geometry, which has been so polished and perfected as the centuries have gone on.—Smith, D. E.

The Teaching of Geometry (Boston, 1911), p. 12.

1847. The culture of the geometric imagination, tending to produce precision in remembrance and invention of visible forms will, therefore, tend directly to increase the appreciation of works of belles-letters.—Hill, Thomas.

The Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 504.

1848.

Yet may we not entirely overlook

The pleasures gathered from the rudiments

Of geometric science. Though advanced

In these inquiries, with regret I speak,

No farther than the threshold, there I found

Both elevation and composed delight:

With Indian awe and wonder, ignorance pleased

With its own struggles, did I meditate

On the relations those abstractions bear

To Nature’s laws.

---

More frequently from the same source I drew

A pleasure quiet and profound, a sense

Of permanent and universal sway,

And paramount belief; there, recognized

A type, for finite natures, of the one

Supreme Existence, the surpassing life

Which to the boundaries of space and time,

Of melancholy space and doleful time,

Superior and incapable of change,

Nor touched by welterings of passion—is,

And hath the name of God. Transcendent peace

And silence did wait upon these thoughts

---

Mighty is the charm

Of those abstractions to a mind beset

With images and haunted by himself,

And specially delightful unto me

Was that clear synthesis built up aloft

So gracefully; even then when it appeared

Not more than a mere plaything, or a toy

To sense embodied: not the thing it is

In verity, an independent world,

Created out of pure intelligence.

—Wordsworth.

The Prelude, Bk. 6.

1849.

’Tis told by one whom stormy waters threw,

With fellow-sufferers by the shipwreck spared,

Upon a desert coast, that having brought

To land a single volume, saved by chance,

A treatise of Geometry, he wont,

Although of food and clothing destitute,

And beyond common wretchedness depressed,

To part from company, and take this book

(Then first a self taught pupil in its truths)

To spots remote, and draw his diagrams

With a long staff upon the sand, and thus

Did oft beguile his sorrow, and almost

Forget his feeling:

—Wordsworth.

The Prelude, Bk. 6.

1850. We study art because we receive pleasure from the great works of the masters, and probably we appreciate them the more because we have dabbled a little in pigments or in clay. We do not expect to be composers, or poets, or sculptors, but we wish to appreciate music and letters and the fine arts, and to derive pleasure from them and be uplifted by them....

So it is with geometry. We study it because we derive pleasure from contact with a great and ancient body of learning that has occupied the attention of master minds during the thousands of years in which it has been perfected, and we are uplifted by it. To deny that our pupils derive this pleasure from the study is to confess ourselves poor teachers, for most pupils do have positive enjoyment in the pursuit of geometry, in spite of the tradition that leads them to proclaim a general dislike for all study. This enjoyment is partly that of the game,—the playing of a game that can always be won, but that cannot be won too easily. It is partly that of the aesthetic, the pleasure of symmetry of form, the delight of fitting things together. But probably it lies chiefly in the mental uplift that geometry brings, the contact with absolute truth, and the approach that one makes to the Infinite. We are not quite sure of any one thing in biology; our knowledge of geology is relatively very slight, and the economic laws of society are uncertain to every one except some individual who attempts to set them forth; but before the world was fashioned the square on the hypotenuse was equal to the sum of the squares on the other two sides of a right triangle, and it will be so after this world is dead; and the inhabitant of Mars, if he exists, probably knows its truth as we know it. The uplift of this contact with absolute truth, with truth eternal, gives pleasure to humanity to a greater or less degree, depending upon the mental equipment of the particular individual; but it probably gives an appreciable amount of pleasure to every student of geometry who has a teacher worthy of the name.—Smith, D. E.

The Teaching of Geometry (Boston, 1911), p. 16.

1851. No other person can judge better of either [the merits of a writer and the merits of his works] than himself; for none have had access to a closer or more deliberate examination of them. It is for this reason, that in proportion that the value of a work is intrinsic, and independent of opinion, the less eagerness will the author feel to conciliate the suffrages of the public. Hence that inward satisfaction, so pure and so complete, which the study of geometry yields. The progress which an individual makes in this science, the degree of eminence which he attains in it, all this may be measured with the same rigorous accuracy as the methods about which his thoughts are employed. It is only when we entertain some doubts about the justness of our own standard, that we become anxious to relieve ourselves from our uncertainty, by comparing it with the standard of another. Now, in all matters which fall under the cognizance of taste, this standard is necessarily somewhat variable; depending on a sort of gross estimate, always a little arbitrary, either in whole or in part; and liable to continual alteration in its dimensions, from negligence, temper, or caprice. In consequence of these circumstances I have no doubt, that if men lived separate from each other, and could in such a situation occupy themselves about anything but self-preservation, they would prefer the study of the exact sciences to the cultivation of the agreeable arts. It is chiefly on account of others, that a man aims at excellence in the latter, it is on his own account that he devotes himself to the former. In a desert island, accordingly, I should think that a poet could scarcely be vain; whereas a geometrician might still enjoy the pride of discovery.—D’Alembert.

Essai sur les Gens Lettres; Melages (Amsterdam 1764), t. 1, p. 334.

1852. If it were required to determine inclined planes of varying inclinations of such lengths that a free rolling body would descend on them in equal times, any one who understands the mechanical laws involved would admit that this would necessitate sundry preparations. But in the circle the proper arrangement takes place of its own accord for an infinite variety of positions yet with the greatest accuracy in each individual case. For all chords which meet the vertical diameter whether at its highest or lowest point, and whatever their inclinations, have this in common: that the free descent along them takes place in equal times. I remember, one bright pupil, who, after I had stated and demonstrated this theorem to him, and he had caught the full import of it, was moved as by a miracle. And, indeed, there is just cause for astonishment and wonder when one beholds such a strange union of manifold things in accordance with such fruitful rules in so plain and simple an object as the circle. Moreover, there is no miracle in nature, which because of its pervading beauty or order, gives greater cause for astonishment, unless it be, for the reason that its causes are not so clearly comprehended, marvel being a daughter of ignorance.—Kant.

Der einzig mÖgliche Beweisgrund zu einer Demonstration des Daseins Gottes; Werke (Hartenstein), Bd. 2, p. 137.

1853. These examples [taken from the geometry of the circle] indicate what a countless number of other such harmonic relations obtain in the properties of space, many of which are manifested in the relations of the various classes of curves in higher geometry, all of which, besides exercising the understanding through intellectual insight, affect the emotion in a similar or even greater degree than the occasional beauties of nature.—Kant.

Der einzig mÖgliche Beweisgrund zu einer Demonstration des Daseins Gottes; Werke (Hartenstein), Bd. 2, p. 138.

1854. But neither thirty years, nor thirty centuries, affect the clearness, or the charm, of Geometrical truths. Such a theorem as “the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides” is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it, and celebrated its advent, it is said, by sacrificing a hecatomb of oxen—a method of doing honor to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even in these degenerate days, marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply of beef.—Dodgson, C. L.

A New Theory of Parallels (London, 1895), Introduction, p. 16.

1855. After Pythagoras discovered his fundamental theorem he sacrificed a hecatomb of oxen. Since that time all dunces10 [Ochsen] tremble whenever a new truth is discovered.—Boerne.

Quoted in Moszkowski: Die unsterbliche Kiste (Berlin, 1908), p. 18.

1856.

—Chamisso, Adelbert von.

Gedichte, 1835 (Haushenbusch), (Berlin, 1889), p. 302.

1857. To the question “Which is the signally most beautiful of geometrical truths?“ Frankland replies: “One star excels another in brightness, but the very sun will be, by common consent, a property of the circle [Euclid, Book 3, Proposition 31] selected for particular mention by Dante, that greatest of all exponents of the beautiful.”—Frankland, W. B.

The Story of Euclid (London, 1902), p. 70.

1858.

—Dante.

Paradise [Carey] Canto 33, lines 122-129.

1859. If geometry were as much opposed to our passions and present interests as is ethics, we should contest it and violate it but little less, notwithstanding all the demonstrations of Euclid and of Archimedes, which you would call dreams and believe full of paralogisms; and Joseph Scaliger, Hobbes, and others, who have written against Euclid and Archimedes, would not find themselves in such a small company as at present.—Leibnitz.

New Essays concerning Human Understanding [Langley], Bk. 1, chap. 2, sect. 12.

1860. I have no fault to find with those who teach geometry. That science is the only one which has not produced sects; it is founded on analysis and on synthesis and on the calculus; it does not occupy itself with probable truth; moreover it has the same method in every country.—Frederick the Great.

Oeuvres (Decker), t. 7, p. 100.

1861. There are, undoubtedly, the most ample reasons for stating both the principles and theorems [of geometry] in their general form,.... But, that an unpractised learner, even in making use of one theorem to demonstrate another, reasons rather from particular to particular than from the general proposition, is manifest from the difficulty he finds in applying a theorem to a case in which the configuration of the diagram is extremely unlike that of the diagram by which the original theorem was demonstrated. A difficulty which, except in cases of unusual mental powers, long practice can alone remove, and removes chiefly by rendering us familiar with all the configurations consistent with the general conditions of the theorem.—Mill, J. S.

System of Logic, Bk. 2, chap. 3, sect. 3.

1862. The reason why I impute any defect to geometry, is, because its original and fundamental principles are deriv’d merely from appearances; and it may perhaps be imagin’d, that this defect must always attend it, and keep it from ever reaching a greater exactness in the comparison of objects or ideas, than what our eye or imagination alone is able to attain. I own that this defect so far attends it, as to keep it from ever aspiring to a full certainty. But since these fundamental principles depend on the easiest and least deceitful appearances, they bestow on their consequences a degree of exactness, of which these consequences are singly incapable.—Hume, D.

A Treatise of Human Nature, Part 3, sect. 1.

1863. I have already observed, that geometry, or the art, by which we fix the proportions of figures, tho’ it much excels both in universality and exactness, the loose judgments of the senses and imagination; yet never attains a perfect precision and exactness. Its first principles are still drawn from the general appearance of the objects; and that appearance can never afford us any security, when we examine the prodigious minuteness of which nature is susceptible....

There remain, therefore, algebra and arithmetic as the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty.—Hume, D.

A Treatise of Human Nature, Part 3, sect. 1.

1864. All geometrical reasoning is, in the last resort, circular: if we start by assuming points, they can only be defined by the lines or planes which relate them; and if we start by assuming lines or planes, they can only be defined by the points through which they pass.—Russell, Bertrand.

Foundations of Geometry (Cambridge, 1897), p. 120.

1865. The description of right lines and circles, upon which Geometry is founded, belongs to Mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.... it requires that the learner should first be taught to describe these accurately, before he enters upon Geometry; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from Mechanics; by Geometry the use of them, when solved, is shown.... Therefore Geometry is founded in mechanical practice, and is nothing but that part of universal Mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that Geometry is commonly referred to their magnitudes, and Mechanics to their motion.—Newton.

Philosophiae Naturalis Principia Mathematica, Praefat.

1866. We must, then, admit ... that there is an independent science of geometry just as there is an independent science of physics, and that either of these may be treated by mathematical methods. Thus geometry becomes the simplest of the natural sciences, and its axioms are of the nature of physical laws, to be tested by experience and to be regarded as true only within the limits of error of observation—BÔcher, Maxime.

Bulletin American Mathematical Society, Vol. 2 (1904), p. 124.

1867. Geometry is not an experimental science; experience forms merely the occasion for our reflecting upon the geometrical ideas which pre-exist in us. But the occasion is necessary, if it did not exist we should not reflect, and if our experiences were different, doubtless our reflections would also be different. Space is not a form of sensibility; it is an instrument which serves us not to represent things to ourselves, but to reason upon things.—PoincarÉ, H.

On the Foundations of Geometry; Monist, Vol. 9 (1898-1899), p. 41.

1868. It has been said that geometry is an instrument. The comparison may be admitted, provided it is granted at the same time that this instrument, like Proteus in the fable, ought constantly to change its form.—Arago.

Oeuvres, t. 2 (1854), p. 694.

1869. It is essential that the treatment [of geometry] should be rid of everything superfluous, for the superfluous is an obstacle to the acquisition of knowledge; it should select everything that embraces the subject and brings it to a focus, for this is of the highest service to science; it must have great regard both to clearness and to conciseness, for their opposites trouble our understanding; it must aim to generalize its theorems, for the division of knowledge into small elements renders it difficult of comprehension.—Proclus.

Quoted in D. E. Smith: The Teaching of Geometry (Boston, 1911), p. 71.

1870. Many are acquainted with mathematics, but mathesis few know. For it is one thing to know a number of propositions and to make some obvious deductions from them, by accident rather than by any sure method of procedure, another thing to know clearly the nature and character of the science itself, to penetrate into its inmost recesses, and to be instructed by its universal principles, by which facility in working out countless problems and their proofs is secured. For as the majority of artists, by copying the same model again and again, gain certain technical skill in painting, but no other knowledge of the art of painting than what their eyes suggest, so many, having read the books of Euclid and other geometricians, are wont to devise, in imitation of them and to prove some propositions, but the most profound method of solving more difficult demonstrations and problems they are utterly ignorant of.—LaFaille, J. C.

Theoremata de Centro Gravitatis (Anvers, 1632), Praefat.

1871. The elements of plane geometry should precede algebra for every reason known to sound educational theory. It is more fundamental, more concrete, and it deals with things and their relations rather than with symbols.—Butler, N. M.

The Meaning of Education etc. (New York, 1905), p. 171.

1872. The reason why geometry is not so difficult as algebra, is to be found in the less general nature of the symbols employed. In algebra a general proposition respecting numbers is to be proved. Letters are taken which may represent any of the numbers in question, and the course of the demonstration, far from making use of a particular case, does not even allow that any reasoning, however general in its nature, is conclusive, unless the symbols are as general as the arguments.... In geometry on the contrary, at least in the elementary parts, any proposition may be safely demonstrated on reasonings on any one particular example.... It also affords some facility that the results of elementary geometry are in many cases sufficiently evident of themselves to the eye; for instance, that two sides of a triangle are greater than the third, whereas in algebra many rudimentary propositions derive no evidence from the senses; for example, that a³-b³ is always divisible without a remainder by a-b.—De Morgan, A.

On the Study and Difficulties of Mathematics (Chicago, 1902), chap. 13.

1873. The principal characteristics of the ancient geometry are:—

(1) A wonderful clearness and definiteness of its concepts and an almost perfect logical rigour of its conclusions.

(2) A complete want of general principles and methods.... In the demonstration of a theorem, there were, for the ancient geometers, as many different cases requiring separate proof as there were different positions of the lines. The greatest geometers considered it necessary to treat all possible cases independently of each other, and to prove each with equal fulness. To devise methods by which all the various cases could all be disposed of with one stroke, was beyond the power of the ancients.—Cajori, F.

History of Mathematics (New York, 1897), p. 62.

1874. It has been observed that the ancient geometers made use of a kind of analysis, which they employed in the solution of problems, although they begrudged to posterity the knowledge of it.—Descartes.

Rules for the Direction of the Mind; The Philosophy of Descartes [Torrey] (New York, 1892), p. 68.

1875. The ancients studied geometry with reference to the bodies under notice, or specially: the moderns study it with reference to the phenomena to be considered, or generally. The ancients extracted all they could out of one line or surface, before passing to another; and each inquiry gave little or no assistance in the next. The moderns, since Descartes, employ themselves on questions which relate to any figure whatever. They abstract, to treat by itself, every question relating to the same geometrical phenomenon, in whatever bodies it may be considered. Geometers can thus rise to the study of new geometrical conceptions, which, applied to the curves investigated by the ancients, have brought out new properties never suspected by them.—Comte.

Positive Philosophy [Martineau], Bk. 1, chap. 3.

1876. It is astonishing that this subject [projective geometry] should be so generally ignored, for mathematics offers nothing more attractive. It possesses the concreteness of the ancient geometry without the tedious particularity, and the power of the analytical geometry without the reckoning, and by the beauty of its ideas and methods illustrates the esthetic generality which is the charm of higher mathematics, but which the elementary mathematics generally lacks.

Report of the Committee of Ten on Secondary School Studies (Chicago, 1894), p. 116.

1877. There exist a small number of very simple fundamental relations which contain the scheme, according to which the remaining mass of theorems [in projective geometry] permit of orderly and easy development.

By a proper appropriation of a few fundamental relations one becomes master of the whole subject; order takes the place of chaos, one beholds how all parts fit naturally into each other, and arrange themselves serially in the most beautiful order, and how related parts combine into well-defined groups. In this manner one arrives, as it were, at the elements, which nature herself employs in order to endow figures with numberless properties with the utmost economy and simplicity.—Steiner, J.

Werke, Bd. 1 (1881), p. 233.

1878. Euclid once said to his king Ptolemy, who, as is easily understood, found the painstaking study of the “Elements” repellant, “There exists no royal road to mathematics.” But we may add: Modern geometry is a royal road. It has disclosed “the organism, by means of which the most heterogeneous phenomena in the world of space are united one with another ” (Steiner), and has, as we may say without exaggeration, almost attained to the scientific ideal.—Hankel, H.

Die Entwickelung der Mathematik in den letzten Jahrhunderten (TÜbingen, 1869).

1879. The two mathematically fundamental things in projective geometry are anharmonic ratio, and the quadrilateral construction. Everything else follows mathematically from these two.—Russell, Bertrand.

Foundations of Geometry (Cambridge, 1897), p. 122.

1880. ... Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint,—an enchanted realm where thought is double and flows throughout in parallel streams.—Keyser, C. J.

Lectures on Science, Philosophy and Arts (New York, 1908), p. 2.

1881. The ancients, in the early days of the science, made great use of the graphic method, even in the form of construction; as when Aristarchus of Samos estimated the distance of the sun and moon from the earth on a triangle constructed as nearly as possible in resemblance to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature, and when therefore an observation of the angle at the earth would define the triangle. Archimedes himself, though he was the first to introduce calculated determinations into geometry, frequently used the same means. The introduction of trigonometry lessened the practice; but did not abolish it. The Greeks and Arabians employed it still for a great number of investigations for which we now consider the use of the Calculus indispensable.—Comte, A.

Positive Philosophy [Martineau], Bk. 1, chap. 3.

1882. A mathematical problem may usually be attacked by what is termed in military parlance the method of “systematic approach;” that is to say, its solution may be gradually felt for, even though the successive steps leading to that solution cannot be clearly foreseen. But a Descriptive Geometry problem must be seen through and through before it can be attempted. The entire scope of its conditions, as well as each step toward its solution, must be grasped by the imagination. It must be “taken by assault”—Clarke, G. S.

Quoted in W. S. Hall: Descriptive Geometry (New York, 1902), chap. 1.

1883. The grand use [of Descriptive Geometry] is in its application to the industrial arts;—its few abstract problems, capable of invariable solution, relating essentially to the contacts and intersections of surfaces; so that all the geometrical questions which may arise in any of the various arts of construction,—as stone-cutting, carpentry, perspective, dialing, fortification, etc.,—can always be treated as simple individual cases of a single theory, the solution being certainly obtainable through the particular circumstances of each case. This creation must be very important in the eyes of philosophers who think that all human achievement, thus far, is only a first step toward a philosophical renovation of the labours of mankind; towards that precision and logical character which can alone ensure the future progression of all arts.... Of Descriptive Geometry, it may further be said that it usefully exercises the student’s faculty of Imagination,—of conceiving of complicated geometrical combinations in space; and that, while it belongs to the geometry of the ancients by the character of its solutions, it approaches to the geometry of the moderns by the nature of the questions which compose it.—Comte, A.

Positive Philosophy [Martineau], Bk. 1, chap. 3.

1884. There is perhaps nothing which so occupies, as it were, the middle position of mathematics, as trigonometry.—Herbart, J. F.

Idee eines ABC der Anschauung; Werke (Kehrbach) (Langensalza, 1890), Bd. 1, p. 174.

1885. Trigonometry contains the science of continually undulating magnitude: meaning magnitude which becomes alternately greater and less, without any termination to succession of increase and decrease.... All trigonometric functions are not undulating: but it may be stated that in common algebra nothing but infinite series undulate: in trigonometry nothing but infinite series do not undulate.—De Morgan, A.

Trigonometry and Double Algebra (London, 1849), Bk. 1, chap. 1.

1886. Sin²f is odious to me, even though Laplace made use of it; should it be feared that sinf² might become ambiguous, which would perhaps never occur, or at most very rarely when speaking of sin (f²), well then, let us write (sinf)², but not sin²f, which by analogy should signify sin(sinf).—Gauss.

Gauss-Schumacher Briefwechsel, Bd. 3, p. 292; Bd. 4, p. 63.

1887. Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical trigonometry.—Tait, P. G.

Encyclopedia Britannica, 9th Edition; Article “Quaternions”

1888. “Napier’s Rule of circular parts” is perhaps the happiest example of artificial memory that is known.—Cajori, F.

History of Mathematics (New York, 1897), p. 165.

1889. The analytical equations, unknown to the ancients, which Descartes first introduced into the study of curves and surfaces, are not restricted to the properties of figures, and to those properties which are the object of rational mechanics; they apply to all phenomena in general. There cannot be a language more universal and more simple, more free from errors and obscurities, that is to say, better adapted to express the invariable relations of nature.—Fourier.

ThÉorie Analytique de la Chaleur, Discours PrÉliminaire.

1890. It is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery—Columbus when he first saw the Western shore, Pizarro when he stared at the Pacific Ocean, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method of co-ordinate geometry.—Whitehead, A. N.

An Introduction to Mathematics (New York, 1911), p. 122.

1891. It is often said that an equation contains only what has been put into it. It is easy to reply that the new form under which things are found often constitutes by itself an important discovery. But there is something more: analysis, by the simple play of its symbols, may suggest generalizations far beyond the original limits.—Picard, E.

Bulletin American Mathematical Society, Vol. 2 (1905), p. 409.

1892. It is not the Simplicity of the Equation, but the Easiness of the Description, which is to determine the Choice of our Lines for the Constructions of Problems. For the Equation that expresses a Parabola is more simple than that that expresses the Circle, and yet the Circle, by its more simple Construction, is admitted before it.—Newton.

The Linear Constructions of Equations; Universal Arithmetic (London, 1769), Vol. 2, p. 468.

1893. The pursuit of mathematics unfolds its formative power completely only with the transition from the elementary subjects to analytical geometry. Unquestionably the simplest geometry and algebra already accustom the mind to sharp quantitative thinking, as also to assume as true only axioms and what has been proven. But the representation of functions by curves or surfaces reveals a new world of concepts and teaches the use of one of the most fruitful methods, which the human mind ever employed to increase its own effectiveness. What the discovery of this method by Vieta and Descartes brought to humanity, that it brings today to every one who is in any measure endowed for such things: a life-epoch-making beam of light [Lichtblick]. This method has its roots in the farthest depths of human cognition and so has an entirely different significance, than the most ingenious artifice which serves a special purpose.—Bois-Reymond, Emil du.

Reden, Bd. 1 (Leipzig, 1885), p. 287.

1894.

—Anonymous.