A History of the Teaching of Elementary Geometry: With Reference to Present-day ProblemsTeachers College, Columbia University, 1906 - 163 strani |
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Zadetki 1–5 od 34
Stran 6
... common use to - day . Piazzi Smyth gives us further information regarding the Great Pyramid at Cheops : ( d ) The angle of the entrance pas- sage on the north is a little over 26 ° . ( e ) The angle of the northern air passage is 33 ...
... common use to - day . Piazzi Smyth gives us further information regarding the Great Pyramid at Cheops : ( d ) The angle of the entrance pas- sage on the north is a little over 26 ° . ( e ) The angle of the northern air passage is 33 ...
Stran 15
... common . Hermotimus of Colophon pursued further the lines opened up by Eudoxus and Theætetus , and discovered many pro- positions of the ' Elements ' and composed some on Loci . Philip- pus of Mende , a pupil of Plato and incited by him ...
... common . Hermotimus of Colophon pursued further the lines opened up by Eudoxus and Theætetus , and discovered many pro- positions of the ' Elements ' and composed some on Loci . Philip- pus of Mende , a pupil of Plato and incited by him ...
Stran 16
... common logical sequence will be referred to in the last chapter of this essay . Educational Features of the Greek Geometry It is not to be expected in the early development of any science that any but mature minds should engage in its ...
... common logical sequence will be referred to in the last chapter of this essay . Educational Features of the Greek Geometry It is not to be expected in the early development of any science that any but mature minds should engage in its ...
Stran 17
... common . That the Greek mind was more interested in the chain of reasoning than in the subject - matter itself is illustrated in some of the dialogues of Plato.2 Under the old Greek education , the youth from sixteen to eighteen ...
... common . That the Greek mind was more interested in the chain of reasoning than in the subject - matter itself is illustrated in some of the dialogues of Plato.2 Under the old Greek education , the youth from sixteen to eighteen ...
Stran 21
... common use for the drawing of figures , for this was a common method among the orientals in doing their calculating , and the Greeks certainly made use of this convenient mode.1 It will be pointed out later that the tendency to hold to ...
... common use for the drawing of figures , for this was a common method among the orientals in doing their calculating , and the Greeks certainly made use of this convenient mode.1 It will be pointed out later that the tendency to hold to ...
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algebra applications Archimedes arithmetic astronomy axiom began Boethius books of Euclid Cantor Chapter circle colleges conic sections constructions course Egyptians eighteenth century elementary geometry Elements employed Erfurt Euclid Euclidean France geome géométrie geometry was taught Gerbert Germany Geschichte given Greeks Günther Gymnasia Gymnasium heights and distances Heron of Alexandria high school Ibid influence institutions isosceles L'enseignement later Latin Legendre Leonardo of Pisa logical geometry lycées Math mathematics mathématiques mensuration ment mentioned method middle Monumenta Germaniæ parallel parallel axiom plane and solid plane figures plane geometry Plato practical geometry problems Proclus Professor proof proportion propositions pupils Pythagoreans secondary schools sequence shows sixteenth century solid geometry sphæra straight line study of geometry subject-matter of geometry surveying teachers teaching of geometry teaching of mathematics text-books texts Thales theorems theory tions to-day treatment triangle trigonometry universities Vormbaum Zittau
Priljubljeni odlomki
Stran 15 - The Comparison of the Five Regular Solids," was written by Aristaaus. 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 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 126 - ... 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 36 - Heron to prove his formula for the area of a triangle in terms of its sides, is...
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 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 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*, to which he applied...
Stran 125 - Geometry : The subjects of Euclid I.-IV. with simple deductions, including easy loci and the areas of triangles and parallelograms of which the bases and altitudes are given commensurable lengths.