"It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters, insofar as they are subject to reasoning, all that we can do in arithmetic and geometry." (Gottfried W Leibniz, 1677)
"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)
"Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)
"I found the elements of a new characteristic, completely different from Algebra and which will have great advantages for the exact and natural mental representation, although without figures, of everything that depends on the imagination. Algebra is nothing but the characteristic of undetermined numbers or magnitudes. But it does not directly express the place, angles and motions, from which it follows that it is often difficult to reduce, in a computation, what is in a figure, and that it is even more difficult to find geometrical proofs and constructions which are enough practical even when the Algebraic calculus is all done." (Gottfried W Leibniz, [letter to Christiaan Huygens] 1679)
"But as I considered the matter carefully it gradually came to light that all those matters only were referred to Mathematics in which order and measurement are investigated investigated, and it makes no difference whether it be in numbers, figures, stars, sounds or any other objects that the question of measurement arises." (René Descartes, "Rules for the Direction of the Mind", 1684)
"These Exponents they call Logarithms, which are Artificial Numbers, so answering to the Natural Numbers, as that the addition and Subtraction of these, answers to the Multiplication and Division of the Natural Numbers. By this means," (the Tables being once made) the Work of Multiplication and Division is performed by Addition and Subtraction; and consequently that of Squaring and Cubing, by Duplication and Triplication; and that of Extracting the Square and Cubic Root, by Bisection and Trisection; and the like in the higher Powers." (John Wallis, "Of Logarithms, Their Invention and Use", 1685)
"These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square" (when they happen,) are reputed to imply that the Case proposed is Impossible. And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible that any Number (Negative or Affirmative) Multiplied into it- self can produce" (for instance) -4. Since that Like Signs (whether + or -) will produce +; and there- fore not -4. But it is also Impossible that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing or any Number Fewer than None. Yet is not that Supposition(of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing. Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were +; but to be interpreted in a contrary sense." (John Wallis, in "Treatise of Algebra", 1685)
"The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc’d to a Mathematical Reasoning; and when they cannot it’s a sign our knowledge of them is very small and confus’d; and when a Mathematical Reasoning can be had it’s as great a folly to make use of any other, as to grope for a thing in the dark, when you have a Candle standing by you." (John Arbuthnot, "Of the Laws of Chance", 1692)
No comments:
Post a Comment