"[…] chance, that is, an infinite number of events, with respect to which our ignorance will not permit us to perceive their causes, and the chain that connects them together. Now, this chance has a greater share in our education than is imagined. It is this that places certain objects before us and, in consequence of this, occasions more happy ideas, and sometimes leads us to the greatest discoveries […]" (Claude A Helvetius, "On Mind", 1751)
"One must admit that it is not a simple matter to accurately outline the idea of negative numbers, and that some capable people have added to the confusion by their inexact pronouncements. To say that the negative numbers are below nothing is to assert an unimaginable thing." (Jean le Rond d'Alembert, "Negatif", Encyclopédie [1751 – 1772])
"[…] the algebraic rules of operation with negative numbers are generally admitted by everyone and acknowledged as exact, whatever idea we may have about this quantities. " (Jean le Rond d'Alembert, Encyclopédie, [1751 – 1772])
"The properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations." (Leonhard Euler, "Specimen de usu observationum in mathesi pura, ["Example of the use of observation in pure mathematics"], cca. 1753)
"The theory of non-extension is also convenient for eliminating from Nature all idea of a coexistent continuum — to explain which philosophers have up till now laboured so very hard & generally in vain. Assuming non-extension, no division of a real entity can be carried on indefinitely ; we shall not be brought to a standstill when we seek to find out whether the number of parts that are actually distinct & separable is finite or infinite ; nor with it will there come in any of those other truly innumerable difficulties that, with the idea of continuous composition, have given so much trouble- to philosophers. For if the primary elements of matter are perfectly non-extended & indivisible points separated from one another by some definite interval, then the number of points in any given mass must bc finite ; because all the distances are finite." (Roger J Boscovich, "Philosophiae Naturalis Theoria Redacta Ad Unicam Legera Virium in Natura Existentium, 1758)
"[negative numbers] darken the very whole doctrines of the equations and to make dark of the things which are in their nature excessively obvious and simple. It would have been desirable in consequence that the negative roots were never allowed in algebra or that they were discarded." (Francis Meseres, 1759)
"We must distinguish carefully the ratios that our ears really perceive from those that the sounds expressed as numbers include." (Leonhard Euler, "Conjecture into the reasons for some dissonances generally heard in music", 1760)
"In order to know the curvature of a curve, the determination of the radius of the osculating circle furnishes us the best measure, where for each point of the curve we find a circle whose curvature is precisely the same. However, when one looks for the curvature of a surface, the question is very equivocal and not at all susceptible to an absolute response, as in the case above. There are only spherical surfaces where one would be able to measure the curvature, assuming the curvature of the sphere is the curvature of its great circles, and whose radius could be considered the appropriate measure. But for other surfaces one doesn’t know even how to compare a surface with a sphere, as when one can always compare the curvature of a curve with that of a circle. The reason is evident, since at each point of a surface there are an infinite number of different curvatures. One has to only consider a cylinder, where along the directions parallel to the axis, there is no curvature, whereas in the directions perpendicular to the axis, which are circles, the curvatures are all the same, and all other oblique sections to the axis give a particular curvature. It’s the same for all other surfaces, where it can happen that in one direction the curvature is convex, and in another it is concave, as in those resembling a saddle." (Leonhard Euler, "Recherches sur la courbure des surfaces", 1767)
"Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such numbers, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in the imagination." (Leonhard Euler, "Vollständige Anleitung zur Algebra", 1768-69)
"All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible." (Leonhard Euler, "Algebra" , 1770)
No comments:
Post a Comment