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11. From any point outside a line two perpendiculars can be constructed to the line.

Let AB be any line and P any point not on
AB, and draw PA, PB.
With PA and PB as diameters construct © A

intersecting AB at Y and X respectively, and
draw PX, PY.

ZPYA = 90°, and ZBXP= 90°.

$ 18, 3 Hence

both PX and PY are I to AB.

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12. The whole of a line is equal to one of its parts.

In this A, CP is I to AB, and CX is drawn so as to make ZACX=ZB.

Then AAXC is similar to A ACB.

Hence these A are proportional to the squares of corresponding sides, and,

A4 since they have equal altitudes, to their

X P bases also.


AB Then

$$ 26, 3; 26, 1 AAXC TX" AX'

BC2CX and hence

$ 21,4
AC? + AB" - 2 ABAP ACP+AX'- 2 AX AP

$ 22, 10


+AB-2 AP= +AX – 2 AP;


AC? whence

- AX= - AB,

AC" - AB. AX AC - AB. AX


$ 21, 2
the whole of a line is equal to one of its parts.
13. Show how to arrange six matches so that each match
shall touch four others.

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268. Ancient Geometry. The geometry of very ancient peoples was largely the mensuration of simple areas and volumes such as is taught to children in elementary arithmetic today. They learned how to find the area of a rectangle, and in the oldest mathematical records that we have there is some discussion of triangles and of the volumes of solids.

Our earliest documents relating to geometry have come to us from Babylon and Egypt. Those from Babylon were written, about 2000 B.C., on small clay tablets (some of them about the size of the hand) which were afterwards baked in the sun. They show that the Babylonians of that period knew something of land measures and perhaps had advanced far enough to compute the area of a trapezoid. For the mensuration of the circle they later used, as did the early Hebrews, the value T = 3.

The first definite knowledge that we have of Egyptian mathematics comes to us from two manuscripts copied on papyrus, a kind of paper used in the countries about the Mediterranean in early times. One of these manuscripts was made by one Aah-mesu (the Moon-born), commonly called Ahmes, who flourished probably about 1600 B.C. The original from which he copied, written about 2000 B.C., has been lost, but the papyrus of Ahmes, written over three thousand years ago, is still preserved and is now in the British Museum. In this manuscript, which is devoted chiefly to fractions and to a crude algebra, is found some work on mensuration. While there is some doubt as to the translation of some of the statements, apparently the curious rules given include the ones that the area of an isosceles triangle is half the product of the base and one of

the equal sides, and that the area of a trapezoid with bases b, b' and nonparallel sides each equal to a is ta(b+b'). One noteworthy advance appears, however, where Ahmes gives a rule for finding the area of a circle, substantially as follows: Multiply the square on the radius by (1,6).


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Part of the Ahmes Papyrus The oldest extensive book on mathematics in the world, a papyrus roll

written by Ahmes about 1550 B.C.

This is equivalent to taking for 7 the value 3.1605, and is the earliest known case of so close an approximation.

The second ancient Egyptian manuscript, which may have antedated slightly the work of Ahmes, is now in Russia. It is on mensuration and apparently contains one interesting case of the mensuration of a solid.

§§ 269,270 GREEK GEOMETRY 229

269. Early Greek Geometry. From Egypt, and possibly from Babylon, geometry passed to the shores of Asia Minor and Greece. The scientific study of the subject begins with Thales, one of the Seven Wise Men of the early Greek civilization. Born at Miletus about 624 B.C., he died there about 548 B.C. He founded at Miletus a school of mathematics and philosophy, known as the Ionic School. How elementary the knowledge of geometry was at that time may be understood from the fact that tra- dition attributes to Thales only about four propositions.

The greatest pupil of Thales, and one of the most remarkable men of antiquity, was Pythagoras, born probably on the island of Samos, just off the coast of Asia Minor, about the year 580 B.C. Pythagoras set A coin of S

coln of Samos, one forth as a young man to travel. He went of the oldest known to Miletus and studied under Thales, prob- Portrait medals of a - mathematician

ably spent several years in Egypt, and very likely went to Babylon. He then founded a school at Crotona, in Italy. He is said to have been the first to demonstrate the proposition in geometry that the square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the other two sides.

270. Euclid. The first great textbook on geometry, and the most famous one that has ever appeared, was written by Euclid, who taught mathematics in the great university at Alexandria, Egypt, about 300 B.C. Alexandria, named in honor of Alexander the Great, was then practically a Greek city, as it was ruled by the Greeks.

Euclid's work is known as the Elements, and, in common with all ancient works, the leading divisions were called “books,” as is seen in the Bible and in the works



of such Latin writers as Cæsar and Vergil. This is why we speak today of the various books of geometry. In this work Euclid placed all the leading propositions of plane geometry that were then known, and arranged them

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predarislimms liber dementozam Euclidis peripi,
cuciffimi:in artam Beometrie incipit quáfocictame: Dc principijs ple notis:epimo se diffini

Unctus et cuius ps nó eft.alinca cft nonibus carondem.
lógitudo finc latitudine culo quidé cs/
tremitates ft duo pucta. Olinca roca

leab vno pucto ad aliū b:euillima eric
fioi extremitates suas vtrúq;cox rcci Dentro
piens. (Supficiese logitudine a lari
ludiné tan bgcuotermi quíde fürlinet.
asupficies plana ć ab vna linca ada,
há atélio i crtremitates suas recipies faphacs plans.

QAngulus planusi duarú lincarù al

cemusytactus:quaf efpafio è lap lup, hcié applicatioq; nó directa. Quádo aut angulum ptinei due linee recte rectilineangnlas noiar. I Ai rccta linea fupreas fteterit duoqz angoli strobiq; fucrit eğles:COK VIETqzrecr


Clincaqz linee supstás ei cuilupftat pendicularis vocat.can
gulus vo qui recto maior e obtufus dicit.Angulovo minoz're
cto acar"appcllat. Terminé qo pniulcourq; finisé. Sigura
è atmino l'ermis ptinct. Circul'c figura plana vna qacm li
nca ptéra: circúferentia noiat:in cui?mcdio pucrc:a quo'ois

linee recte ad arcúfereria ereutes fibiiuicez sur equalcs. Erbic
quidé pucro cótrú cirauli di. Q Diameter airculic linea recra que
tupci'cente tráliens extremitatcsq; fuas circúferétie applicans
circulu i duo media diuidit a scmiaralus é figura plana dia

Porno malo
metro circulia mcdietate arcüfcrentic ptena. Porno circa
lić figura plana recta linea a parte arcúferitie ptcra: 1emicircu,
lo quidé ant maioz aut minor. O hectilinee figure fürörccnslu
ncis corineni quarú quedá milatere amb rectus lincis: qucds


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First Page of Euclid's Elements From the first printed edition, Venice, 1482

en claro
quadrilatere a quoz rechslineis. (da mltilaterc que plunbus
43 quatuor rectis lineis continení. i Sigurarú frilarcrarú:alia

cit triangulus bństria latcta cquala. Aua miangulus duo bris
cólia latera. Alia triangulus triù incqualiam latcrú, by iteru

Ongono ambos ombusontbe
alia cft odbogoniú:vnú. rectuni angulun babcns. Alia i am
bligoruum aliquem obtulum angulum babene. Alia eft ongom
um:in qua tres asiguli funt acon asigurarú aute quadrilateraf I etrasa'lis! Wrare
Alia cft drarum quod eft cquilateru atq; rectangulú. Alaeit
retragon'longo:gcft figura rectangula :icd equilatcra non eft.
Alia cf belmuaym: que el cquilatera : led rectangula non ch.


Do cquales lorca


in a logical order. Most geometries of any importance since his time have been based upon this great work of Euclid, and improvements in the sequence, symbols, and wording have been made as occasion demanded.

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