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I did not admire Euclid, who rather seeks a chain of demonstration than a connection of ideas: I preferred the geometry of Father Lama, who from that time became one of my favorite authors, and whose works I yet read with pleasure.
* This is intelligible without calculation but only for the two-dimensional case if we revert once more to the case of the disc on the surface of the sphere. In this way, by using as stepping-stones the practice in thinking and visualisation which Euclidean geometry gives us, we have acquired a mental picture of spherical geometry.
That this can be done with a certain degree of approximation is fairly evident If I see three people A, B, and C sitting in a row, I become aware of the fact which may be expressed by saying that B is between A and C rather than that A is between B and C, or C is between A and B. This relation of "between" which is thus perceived to hold has some of the abstract logical properties of those three-term relations which, we saw, give rise to a geometry, but its properties fail to be exact, and are not, as empirically given, amenable to the kind of treatment at which geometry aims.
Neither is there any hint as to the manner of learning Arithmetic and the Elements of Geometry, save that the latter might be picked up "even playing, as the old manner was." On another part of the training of this First Class, however, Milton is more specific.
Now mind, what I am to prove is this: if any triangle has two sides equal, the angles opposite those sides are also equal." "And what difference does it make if they're equal or not?" said one of the men who stood near Kathleen. "Be still there," the King said; "do we want to make telephones or do we not? And sure we can't make telephones without geometry. Hasn't Terence told you that?"
Perfect lines and surfaces do not exist within the region of our experience; yet the conclusions of geometry are none the less true ideally, though in any particular concrete instance they are only approximately realized. Just so with the conception of a frictionless fluid.
They had never so much as heard of the names of any of those philosophers that are so famous in these parts of the world, before we went among them; and yet they had made the same discoveries as the Greeks, both in music, logic, arithmetic, and geometry.
The Milesians, then, had formed the conception of an eternal matter out of which all things are produced and into which all things return, and the conception of Matter belongs to philosophy rather than to science. But besides this they had laid the foundations of geometry, and that led in other hands to the formulation of the correlative conception of Limit or Form.
As we have seen, the discovery of the incommensurable rendered inadequate the Pythagorean theory of proportion, which applied to commensurable magnitudes only. It would no doubt be possible, in most cases, to replace proofs depending on proportions by others; but this involved great inconvenience, and a slur was cast on geometry generally.
In the first part it is set down that it is not blameworthy if one learns grammar and logic in order to distinguish the true and the false. In the second part which begins with "Geometry and Arithmetic" it is set down that the knowledges of the quadrivium have a truth of their own. But they are not the knowledges of piety, and are not to be so applied.
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