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.
I am far from placing logicians by the side of geometers who teach the true way to guide the reason.
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.
The logicians profess to lead the way, the geometers alone reach it, and aside from their science there is no true demonstration.
Every body studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone.
One of the most curious of these cases [geometrical paradoxers] was that of a student, I am not sure but a graduate, of the University of Virginia, who claimed that geometers were in error in assuming that a line had no thickness.
The greatest geometers considered it necessary to treat all possible cases independently of each other, and to prove each with equal fulness.
Some, indeed, have only fourteen; while these geometers are driven to adopt the attitudes they do because they have to shuffle along as best they can, on no more than ten legs altogether.
The curious movements of thesegeometers are due to their comparatively scant supply of legs.
The moths of the large class known asGeometers are so called because the caterpillars, as they loop themselves along, have the air of measuring the space they traverse, as a man might span it with his hand.
All geometers would be men of acuteness if they had sufficient insight, for they never reason falsely on the principles recognised by them.
All fine or acute spirits would be geometers if they could fix their thoughts on the unwonted principles of geometry.
In consequence of this appeal, Mr. Lawson was speedily in correspondence with several of the most able geometers then living, and amongst the rest, Messrs.
It consists of results arrived at by geometers in seeking a proof of the parallel-postulate.
It is the view which geometers have taken of space in general that has made the fourth dimension possible, and not only the fourth, but dimensions of all degrees.
His conception of space, therefore, must have had a profound influence upon the mathematic thought of the day causing it to undergo a rapid reconstruction at the hands of geometers who came after him.
In this way geometers have determined that our space is tridimensional; but it is obvious that this conclusion is based not upon any examination of space itself but upon the measurement of bodies in space.
Geometers claim that space is a system of coordinates necessary for the establishment of a point-position in it.
The utmost ingenuity of geometers failed to combine them satisfactorily with the later Uranian places, and it became evident, either that they were widely erroneous, or that the revolving body was wandering from its ancient track.
But geometers are men; and the contagion of patriotic fervour which swept over Germany after the battle of Leipsic did not spare Gauss's promising pupil.
Geometers have usually preferred to define parallel lines by the property of being in the same plane and never meeting.
Geometers have, in all ages, been open to the imputation of endeavoring to prove the most general facts of the outward world by sophistical reasoning, in order to avoid appeals to the senses.
Gifted, beyond all the sons of men, with the most exquisite perception of form, both physical and metaphysical, they could become geometers and logicians as they became sculptors and artists; beyond that they could hardly rise.
Physical sages there were; but they were geometers and mathematicians, rather than astronomic observers and inquirers.
The logicians profess to show the way, but the geometers alone ever reach it, and aside from their science there is no genuine demonstration.
As far as possible, Euclid and all other good geometers avoid the proof by superposition.
I admire particularly the art with which you bring under uniform methods the divers conclusions scattered among the works ofgeometers and reached by methods entirely different.
But, limited as these elements are, geometershave nevertheless succeeded in treating, in a truly admirable manner, a very great number of important questions, as we shall find in the course of the volume.
We can now form a definite, and, at the same time, sufficiently extended idea of what geometers understand by a veritable equation.
It is in this way, indeed, that the first lines which we may regard as having been truly invented by geometers were obtained, since nature gave directly the straight line and the circle.
But by what right do we consider as equal these two figures which the Euclidean geometers call two circles with the same radius?
There are long chains of theorems where absolute logic has reigned from the very first and, so to speak, quite naturally, where the first geometers have given us models we should constantly imitate and admire.
The study of these properties is the object of a science which has been cultivated by many great geometers and in particular by Riemann and Betti and which has received the name of analysis situs.
To justify this definition it is proper to see whether it is in this way that geometers introduce the notion of three dimensions at the beginning of their works.
These propositions must rest on premises the geometers admit without enunciation.
Shall it be said that we geometers should limit ourselves to awaiting orders, and, in place of cultivating our science for our own delectation, try only to accommodate ourselves to the wants of our patrons?
This is a whole body of doctrine which has attracted the attention of the greatest geometers and where we see flow one from another a series of remarkable theorems.
Among the German geometers of this century, two names above all are illustrious, those of the two scientists who founded the general theory of functions, Weierstrass and Riemann.
Hermite, for example, whom I have just cited, can not be classed among the geometers who make use of the sensible intuition; but neither is he a logician, properly so called.
On the other hand, many geometers believe we can reduce mathematics to the rules of formal logic.
Greek geometers who studied them soon replaced by a proper definition in plano like that for the circle, viz.
The Greek geometers invented other curves; in particular, the conchoid (q.
Such is the cycloid, first conceived by Galileo, and a stumbling-block and cause of contention among geometers long after he had left it, together with his system of the universe, undetermined.
The greatest geometers of the day, Pascal, Roberval, and others, unhesitatingly adopted this method, and employed it in the abstruse researches which engaged their attention.
Many geometers have constructed systems, which they intended to be, and which, with sufficient care in interpretation, really are, free from metrical presuppositions.
Montucla is hard on his compatriot, who, he says, was only saved from the laughter of geometers by his obscurity.
You will throw in some irrelevant questions with a view to lengthen the procedure, like fallacious geometers who complicate a diagram by drawing unnecessary lines.
And you call this habit and practice of the Geometers and others by the name Conception, not Intuition[344]; taking Conception to be something between Opinion on the one side, and Intuitive Insight on the other.
Geometers indeed speak of their visible diagrams, as if their problems were certain practical processes; to erect a perpendicular; to construct a square: and the like.
Not that it varies with the radius; the geometers are right enough on that point: but it varies with the time, in a manner depending upon the difference of the true longitudes of the Sun and Moon.
It reminds us of the old days when real geometers used to think it worth while seriously to demolish pretenders.
The poetic idea seems to be that the geometers try to make a square circle.
The language used by the two great geometers illustrates what I have said: a supreme and guiding intelligence--apart from a blind rule called nature of things--was an hypothesis.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "geometers" in a variety of sentences. We hope that you will now be able to make sentences using this word.
