"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 master keys 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." (Thomas Hill, "The Imagination in Mathematics", North American Review Vol. 85, 1857)
"The language of mathematics, permitting great sharpness and accuracy of de?nition, conduces largely to their power of drawing necessary conclusions. Language is not only a means of recording the results of our thinking; it is an instrument of thought, and that of the highest value." (Thomas Hill, "The Imagination in Mathematics", The North American Review Vol. 85 (176), 1857)
"The prominent reason why a mathematician can be judged by none but mathematicians, is that he uses a peculiar language. The language of mathesis is special and untranslatable. In its simplest forms it can be translated, as, for instance, we say a right angle to mean a square corner. But you go a little higher in the science of mathematics, and it is impossible to dispense with a peculiar language." (Thomas Hill, "The Imagination in Mathematics", The North American Review Vol. 85 (176), 1857)
"By reducing imagination to the power of forming images, and by insisting that no image can be formed except out of the elements furnished by experience, I do not mean to confound imagination with memory; indeed, the frequent occurrence of great strength of memory with comparative feebleness of imagination, would suffice to warn us against such a conclusion." (George H Lewes, "The Principles of Success in Literature", 1865)
"You cannot crown the edifice by this abstraction. The scientific imagination, which is here authoritative, demands as the origin and cause of a series of ether waves a particle of vibrating matter quite as definite, though it may be excessively minute, as that which gives origin to a musical sound. Such a particle we name an atom or a molecule. I think the imagination when focused so as to give definition without penumbral haze, is sure to realise this image at last." (John Tyndall, "The Scientific Use of the Imagination", 1870)
"Accurate and minute measurement seems to the nonscientific imagination a less lofty and dignified work than looking for something new. But nearly all the grandest discoveries of science have been but the rewards of accurate measurement and patient long contained labor in the minute sifting of numerical results." (William T Kelvin, "Report of the British Association For the Advancement of Science" Vol. 41, 1871)
"The imagination is one of the highest prerogatives of man. By this faculty he unites, independently of the will, former images and ideas, and thus creates brilliant and novel results […] The value of the products of our imagination depends of course on the number, accuracy, and clearness of our impressions; on our judgment and taste in selecting or rejecting the involuntary combinations, and to a certain extent on our power of voluntarily combining them." (Charles Darwin, "The Descent of Man", 1874)
"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." (Thomas Hill, "Uses of Mathesis", Bibliotheca Sacra Vol. 32, 1875)
"It would be an error to suppose that the great discoverer seizes at once upon the truth, or has any unerring method of divining it. In all probability the errors of the great mind exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among the first requisites of discovery; but the erroneous guesses must be many times as numerous as those that prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no record remain of more than the hundredth part. […] The truest theories involve suppositions which are inconceivable, and no limit can really be placed to the freedom of hypotheses." (W Stanley Jevons, "The Principles of Science: A Treatise on Logic and Scientific Method", 1877)
"The very genius of the common geometry as a method of reasoning - a system of investigation - is, that it is but a series of observations. The figure being before the eye in actual representation, or before the mind in conception, is so closely scrutinized, that all its distinctive features are perceived; auxiliary lines are drawn (the imagination leading in this), and a new series of inspections is made; and thus, by means of direct, simple observations, the investigation proceeds." (Edward Olney, "Mathematics", The Cyclopedia of Education, 1877)
"But I thoroughly believe myself, and hope to prove to you, that science is full of beautiful pictures, of real poetry, and of wonder-working fairies; and what is more […] though they themselves will always remain invisible, yet you will see their wonderful power at work everywhere around you. […] There is only one gift we must have before we can learn to know them - we must have imagination. I do not mean mere fancy, which creates unreal images and impossible monsters, but imagination, the power of making pictures or images in our mind, of that which is, though it is invisible to us." (Arabella Buckley, Fairyland, 1879)
"[…] deduction consists in constructing an icon or diagram the relations of whose parts shall present a complete analogy with those of the parts of the object of reasoning, of experimenting upon this image in the imagination, and of observing the result so as to discover unnoticed and hidden relations among the parts." (Charles S Peirce, 1885)
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