A History of the Teaching of Elementary Geometry: With Reference to Present-day ProblemsTeachers College, Columbia University, 1906 - 163 strani |
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Stran iii
... side . Among the works that 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 ...
... side . Among the works that 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 ...
Stran 1
... side alone was recognized in pre - grecian geometry . 1 Where contradictions do not arise the term geometry will generally be employed for elementary geometry . I ( b ) That geometry as a pure science ( LIBRARY OF THE.
... side alone was recognized in pre - grecian geometry . 1 Where contradictions do not arise the term geometry will generally be employed for elementary geometry . I ( b ) That geometry as a pure science ( LIBRARY OF THE.
Stran 2
... side of a triangle is less than the sum of the other two . These facts of nature are used because they are serviceable . The Indian fastened his pony to a stake , enabling him to graze in a circle . He knew that the longer the rope ...
... side of a triangle is less than the sum of the other two . These facts of nature are used because they are serviceable . The Indian fastened his pony to a stake , enabling him to graze in a circle . He knew that the longer the rope ...
Stran 5
... sides of the triangle formed were in the ratio of 3:45 . This statement of Democritus tells us that about 420 B.C. the rope - stretchers had fame as practical geometers in Egypt . The pyramids were built during the fourth dynasty ...
... sides of the triangle formed were in the ratio of 3:45 . This statement of Democritus tells us that about 420 B.C. the rope - stretchers had fame as practical geometers in Egypt . The pyramids were built during the fourth dynasty ...
Stran 6
... sides are in the ratio 3 : 45.1 It is hardly necessary to add that the above method of erecting perpendiculars is in common use to - day . Piazzi Smyth gives us further information regarding the Great Pyramid at Cheops : ( d ) The angle ...
... sides are in the ratio 3 : 45.1 It is hardly necessary to add that the above method of erecting perpendiculars is in common use to - day . Piazzi Smyth gives us further information regarding the Great Pyramid at Cheops : ( d ) The angle ...
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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.