A History of the Teaching of Elementary Geometry: With Reference to Present-day Problems

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Teachers College, Columbia University, 1909 - 163 strani
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Stran 102 - But the schools vary in this respect. At the St. Paul's Preparatory School in London the boy at fourteen goes to the Higher School, having done only Euclid I.5 The teaching of geometry in the secondary schools will now be considered. It was not until after the middle of the last century that Oxford and Cambridge began to influence the teaching of geometry in the secondary schools through their local examination systems. Previous to this time and after about 1829, the schools were teaching Euclid...
Stran 28 - Two intersecting straight lines cannot both be parallel to the same straight line. 2. Only one straight line can be drawn through a given point parallel to a given straight line.
Stran 106 - ... abstract reasoning at a more advanced point? Where would be the harm in letting a boy assume the truth of many propositions of the first four books of Euclid, letting him accept their truth partly by faith, partly by trial — giving him the whole fifth book of Euclid by simple algebra; letting him assume the sixth book to be axiomatic; letting him, in fact, begin his severer studies where he is now in the habit of leaving off.
Stran 41 - But geometry has a still greater connection with the art of oratory. Order, in the first place, is necessary in geometry, and is it not also necessary in eloquence? Geometry proves what follows from what precedes, what is unknown from what is known, and do we not draw similar conclusions in speaking?
Stran 15 - The Comparison of the Five Regular Solids," was written by Aristteus. This contained the theorem, "The same circle circumscribes the pentagon of the dodecahedron and the triangle of the icosahedron, these solids being inscribed in the same sphere.
Stran 10 - When is a straight line said to be ' placed in a circle ' ? 2. The angles at the base of an isosceles triangle are equal to...
Stran 28 - ... concerned with teaching than with learning, at all times. No doubt some of the geometries still teach as a self-evident truth the proposition that if two straight lines in one plane meet a third straight line so as to make the sum of the internal angles on one side less than two right angles those two lines will meet on that side if sufficiently prolonged.
Stran 24 - Proclus) invented this method of ex haustions, which may be considered as contained in two propositions. I. If from A more than its half be taken, and from the remainder more than its half, and so on, the remainder will at last become less than B, where B is any magnitude named at the outset (and of the same kind as A), however small. This proposition may be easily proved, and is equally true if the proportion abstracted each time be half or less than half.
Stran 40 - BC ao a radius draw a circle. Let CD be the position of the shadow in the afternoon when its extremity just touches the circumference. BC = CD. Join B, D and draw the perpendicular bisector of BD. This is the required meridian line. 'Cantor I, p. 513ff; Giinther, Geschichte des mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525, p. 115. Hereafter referred to as Giinther. The "Codex" was discovered in 980 by Gerbert, who became Pope Silvester II.
Stran 14 - Elements' more carefully designed, both in the number and the utility of its proofs, and he invented, also, a diorismus (or test for determining) when the proposed problem is possible and when impossible. Eudoxus of Cnidus, a little later than Leon, and a student of the Platonic school, first increased the number of general theorems, added to the three proportions three more, and raised to a considerable quantity the learning, begun by Plato, on the subject of the (golden) section,5 to which he applied...

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