On Geometry (1600-1699)

"Where there is matter, there is geometry." (Johannes Kepler, "Concerning the More Certain Fundamentals of Astrology", 1601)

"Geometry is one and eternal shining in the mind of God." (Johannes Kepler, "Conversation with the Sidereal Messenger" [an open letter to Galileo Galilei] 1610)

"Thus the analysis of angular sections involves geometric and arithmetic secrets which hitherto have been penetrated by no one." (François Viète, cca 1615)

"Mathematic is either Pure or Mixed: To Pure Mathematic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered. [...] Mixed Mathematic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to explain, demonstrate and actuate these." (Francis Bacon, "De Augmentis", 1623)

"Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth." (Galileo Galilei, “The Assayer”, 1623)

"And having thus passed the principles of arithmetic, geometry, astronomy, and geography, with a general compact of physics, they may descend in mathematics to the instrumental science of trigonometry, and from thence to fortification, architecture, engineering, or navigation. And in natural philosophy they may proceed leisurely from the history of meteors, minerals, plants, and living creatures, as far as anatomy. Then also in course might be read to them out of some not tedious writer the institution of physic. […] To set forward all these proceedings in nature and mathematics, what hinders but that they may procure, as oft as shall be needful, the helpful experiences of hunters, fowlers, fishermen, shepherds, gardeners, apothecaries; and in other sciences, architects, engineers, mariners, anatomists." (John Milton, "On Education", 1644)

"In geometry, to be sure, the accident of a fraction or an irrational does not usually prevent equations from being solved readily. nor does the imperfection of a negative, for the subject on which the geometer works is always certain. But multiplicity of affections8 is a hindrance, and the higher the power and the order of an affection, the more likely it is that a fraction or surd9 will appear in the solution of a problem." (François Viète, "On the Structure of Equations as Shown by Zetetics, Plasmatic Modification and Syncrisis", 1646)

"Indeed, many geometric things can be discovered or elucidated by algebraic principles, and yet it does not follow that algebra is geometrical, or even that it is based on geometric principles (as some would seem to think). This close affinity of arithmetic and geometry comes about, rather, because geometry is, as it were, subordinate to arithmetic, and applies universal principles of arithmetic to its special objects." (John Wallis, "Mathesis Universalis", 1657)

"For, Mathematical Demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick, are the only Truths, that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth, according as their Subjects are more or less capable of Mathematical Demonstration." (Christopher Wren, [lecture at Gresham College] 1657)

"Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum's composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysics except he who has passed through this [labyrinth]." (Gottfried W Leibniz, "Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae", 1676)

"After all the progress I have made in these matters, I am still not happy with Algebra, because it provides neither the shortest ways nor the most beautiful constructions of Geometry. This is why when it comes to that, I think that we need another analysis which is properly geometric or linear, which expresses to us directly situm, in the same way as algebra expresses magnitudinem. And I think that I have the tools for that, and that we might represent figures and even engines and motion in character, in the same way as algebra represents numbers in magnitude." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)

"The Excellence of Modern Geometry is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all possible Cases in one view, and in one general Theorem many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the Ancients, might well become the subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases." (Edmund Halley, "An Instance of the Excellence of Modern Algebra in the resolution of the problem of finding the foci of optic glasses universally", Philosophical Transactions, 1694)