A History of the Teaching of Elementary Geometry: With Reference to Present-day ProblemsTeachers College, Columbia University, 1909 - 163 strani |
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Stran iii
... treat the general history of mathematics , those of Cantor , Hankel , Gow , and Allman have been of great assistance . Much information has also been gained from miscellaneous articles , standard works on the history of educa- tion ...
... treat the general history of mathematics , those of Cantor , Hankel , Gow , and Allman have been of great assistance . Much information has also been gained from miscellaneous articles , standard works on the history of educa- tion ...
Stran 1
... treated first chronologically , bringing it down to the present time . Certain modern problems will then be con- sidered in the light of the foregoing historic material . Geometry as studied in the schools to - day has two values : ( a ) ...
... treated first chronologically , bringing it down to the present time . Certain modern problems will then be con- sidered in the light of the foregoing historic material . Geometry as studied in the schools to - day has two values : ( a ) ...
Stran 9
... treated with similar timidity . " In brief , the Egyptians knew how to calculate the areas of some of the simple rectilineal figures , using some rules , how- ever , that were erroneous . Also , they found the capacity of barns by ...
... treated with similar timidity . " In brief , the Egyptians knew how to calculate the areas of some of the simple rectilineal figures , using some rules , how- ever , that were erroneous . Also , they found the capacity of barns by ...
Stran 13
... treat it as a liberal science . Thus far we have the growth of geometry - first , the practical stage as with the Egyptians ... treated as a liberal science . But we have no proof that the subject- matter was yet organized into any fixed ...
... treat it as a liberal science . Thus far we have the growth of geometry - first , the practical stage as with the Egyptians ... treated as a liberal science . But we have no proof that the subject- matter was yet organized into any fixed ...
Stran 16
... treating the sequence of Euclid ; but may be illustrated briefly at this point . The subject - matter of the third book of Euclid , which treats of circles , was developed in the Athenian period . The theory of proportion which follows ...
... treating the sequence of Euclid ; but may be illustrated briefly at this point . The subject - matter of the third book of Euclid , which treats of circles , was developed in the Athenian period . The theory of proportion which follows ...
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algebra angles applications Archimedes areas arithmetic astronomy Bobynin Boethius books of Euclid Cantor Chapter circle colleges conic sections edition Egyptians eighteenth century elementary geometry Elements employed etry Euclid Euclidean Euclidean geometry France geom géométrie geometry is begun geometry was taught Gerbert Germany Geschichte given Greek Günther Gymnasia Gymnasium heights and distances Heron of Alexandria high school higher classes Ibid influence institutions later learning Legendre Leonardo of Pisa logical geometry lycées Math matics mensuration method middle nineteenth century parallel plane and solid plane figures plane geometry Plato practical geometry prepared problems Proclus Professor proof proportion propositions Public Schools pupils Realgymnasium reductio ad absurdum secondary schools sequence sixteenth century solid geometry straight line study of geometry subject-matter of geometry surveying teachers teaching of geometry teaching of mathematics text-book texts Thales theorems theory tion to-day triangle trigonometry universities
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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...