

II 



LESSONS IN GBOMETBY.. 



.. hi.-h coin- k wordu, yj 



/itrpeiv, to uicoMiire (]>ronomieed J//MJC, i 

 , literally sii/niiie- / , ,\ und wan origini 



I to the practical purposo which it* name 

 



in, \\iili t 11 of MOXOK, whoao workii 



I'ivrS till' ( >rif.'ill : 



was informed by tin- priests at Thelics, i 



'.rilnitidii ill' lli. nil Li 



on cqnol portion nf lain!, in tin 

 if u nuadraM^le, 'ind that from ti: a allot) 



roe, I'.v na i.iin tiix. In 



oases, ho\v. . i part <>f the hind was \va-hr, 1 away by 



a mil inline! : Nil.-, tin- iin.|.netnr | 



.im.-elf before the kin/, and >i "iiify what had hap- 

 The kiiur thru useil to send proper ullicer.-i t" 



-certain, by admeasurement, how much of tho land hu< 

 iieeu washed away, in order that tho amount of tho tax to bo 

 paid for tho future mi^'ht lie proportional to the land whicl 



:>cd. From this circumstance I am of opinion thai 

 geometry derived its origin ; and from hence it wan trausmittec 



:vece." Tho existence of tho pyramids, tho ruins of tho 

 temples, and the other architectural remains of ancient Egypt, 

 supply evidence that its inhabitants possessed some knowledgi 



aotry, even in tho higher sense in which wo now use the 

 term ; although it is possible that the geometrical properties ol 

 figures necessary far the construction of such works might have 



Lnown only in tho form of practical rules, without any 

 scientific arrangement of geometrical truths, such as are pre- 

 sented to us in the Elements of Euclid. 



The word " geometry," used in its highest and most extensive 

 meaning, signifies the science of space; or ..hat science which 

 investigates and treats of the properties of, and relations 



:ir among, definite portions of' space, under the abstract 

 division of lines, angles, surfaces, and volumes, without any 

 regard to the physical properties of tho bodies to which they 

 belong. In this sense, it appears to be very doubtful whether 

 the Egyptians or Chaldeans knew anything of the science. It 

 is to the Greeks, therefore, that we must look for the real 

 origin of geometry, as an abstract science. Thales, the Greek 

 philosopher, born 040 B.C., is reported, by ancient historians, 

 to have astonished even the Egyptians by his knowledge of 

 this science. Tho founder of scientific geometry in Greece, 

 however, appears to have been Pythagoras, who was born about 

 568 B.a He discovered tho celebrated 47th proposition of tho 

 first book of Euclid's Elements, and various other valuable and 

 important theorems. He was great also in astronomy, having 

 anticipated the Copornican system of the universe. Plato, 

 another great geometrician, and founder of tho academy at 

 Athens, who was born 429 B.C., was the first who made some 

 advances into what is called the higher geometry. The next 

 uame super-eminent in the science of geometry is that of 

 Euclid, whose " Elements" has been tho principal text-book for 

 learners during a period of more than 2,000 years. lie 

 nourished at Alexandria, in Egypt, about 300 B.C., during the 

 reign of Ptolemy Lagus, who was one of his pupils, and to 

 whom ho made the celebrated reply, when asked if thc-r. 

 shorter way to geometry than by studying his Elements: " No, 



i hen* is no royal road to geometry." 



The prince of ancient mathematicians, however, was the 

 celebrated Archimedes, born at Syracuse B.C. 287, about the 

 period of tho death of Euclid. His discoveries in geometry, 



iiics, and hydrostatics form a remarkable era in the 



of the matiie/natical sciences; and e\en tho remains of 



rks which are still extant constitute tho most valuable 

 part of the ancient (Iret-lc geometry. He was tho first who 

 Jittc-mpted to solve tho celebrated problem of tho rectification of 



'< that is, finding a atnti-iht I'm,' ,'j,t,-tlij r./i/al to the cir- 

 cumference. He found out tho beautiful ratios of tho cylinder 

 to its inscribed sphere and cone, and the quadrature of one of 

 the conic sections. His -iiscovcrics in physics, or natural 

 philosophy, are simple, true, and beautiful. The story of the 

 determination of the specific gravity of the golden crown of his 

 cousin, Hiero, King of Syracuse, is well known ; and the very 

 natural about of " Etipjjiea, evpi;a " (pronounced /' 

 have found it, I have found it ! oil coming out of tho bath, has 



btcorae " household word." Scarcely lot* celebrated was UM 



Forgo, in PampUylia, who *""mhid 



> the reign of Ptolemy 



Etiorgeto*, another king of tho name Ptolomaio dynasty, and 

 hi* contemporaries the " Great Geometer." 

 He wrote wreral book*, f .vork-s, on the ttgbr 



geometry, ^ and greatly extended the domain* of th p'tn" 

 geometry." Other gooin '.ininimoo arose in UM aobool 







genital to happier time*. Claudim I'tol. -utu'iut, the author of 

 it work on astronomy called H>- ..,, the Great 



or Almagest; Pappus, tho nut. 



including Thoon ai 



dfiuj'ht" r Jlyj.iitia period when the *con<f 



Alexandrian library* wan burnt by command of the M^KM]. 



inedan barbarian, the Saracen Calip ; , 040, and 



truing of ages were irrevocably dentroyod. 



Tho dark op; . .1, and little wan done in the adranoe- 



eo until the glorious invention of printing, and the 



general revival of literature about tho middle of the fifteenth 



century. 



Tho ancient Greek geometry was speedily made known to the 

 . through tho in -ilium of translations of, and commen- 

 taries upon, : -i of tho great marten. The Element* 

 of Euclid, indeed, were reckoned BO perfect, that no attempt 

 was made to supersede them ; and tho only object of writer* on 

 geometry in general waa to explain bi works, and to mako 

 what additions they could to the science, in tho same masterly 

 style of composition. A host of names of cmi:. 

 might be mentioned, who succeeded in establishing tie 

 geometry, and in extending its domains. The j 

 however, was Dr. Robert Sim -sor of Mathematics in 

 tho University of Glasgow, who flourished in tho middle- 

 last century. His grand endeavour was to present to modern 

 Europe the Elements of Euclid as they originally appeared in 

 ancient Greece. In this he succeeded to admiration, and his 

 edition of this great work maintains its reputation to the 

 present moment. 



In giving our first lessons on geometry, wo think it advisable 

 to follow what seems to have been the natural cour 

 in the history of this science. The present advanced state of 

 our geometrical knowledge was preceded in early times by a 

 species of practical geometry gathered from experience, and 

 suited to the wants of those who required its application, before 

 any attempt was made to enter very deeply into the study of 

 ;ho theory. The latter was left to the schools of the philoso- 

 phers and the academy of Plato. Accordingly, wo shall precede 

 our disquisitions on the Elements of Euclid and other geometers, 

 joth ancient and modern, by a short system of practical rules 

 ind i a<y explanations in this important science ; and we shall 

 endeavour to make the subject both simple and clear by plain 

 definitions, suitable diagrams, and palpable demonstrations, 

 after the manner of tho French writers on thin subject, who 

 have even in their more elaborate treatises to a great extent 

 abandoned the system of Euclid. 



DEFINITIONS. 



1 . Extension, or the space which any body in nature occupies, 

 las three dimensions, viz., length, breadth, and thi 

 This is Euclid's definition of a geometrical s< 



_'. A point is the beginning of extension, but no part of it; 

 it is said to have position in space, but no magnitude. 



3. A line is extension in one direction only ; hence, it is said 

 jo have length without breadth. Hence, also, the extr< : 



if a line are points ; and lines intersect or cross each other 

 tnly in points. 



4. A ftmi'jlit line is said, by Euclid, to bo that whi. ; 

 venly between its extreme points ; and, by Archimedes, to be 

 ho s/io/v -h of these 

 lefinitions are defective; tho d A straight 

 ine is such, that if any two points be taken in it, the part 

 vhich they intercept (or which lies 1 >m) is the 

 hortest line that can bo drawn between those points. 



5. A crooked line is one composed of straight lines joined at 



The first library, which was founded by Ptolemy Sotor. and which 

 was said to have contained 400,000 manuscript*, waa accidentally 

 unit 47 B.C., when Alexandria was taken by Julias CtosaT. The 

 ccond library * supposed to hare contained 700,000 volumes. 



