KNOWLEDGE IMPLIED BY EARLY ASTRONOMY. 167 



in other sciences accompanied, and was necessary to, these 

 astronomical previsions. In the first place, there must 

 clearly have been a tolerably efficient system of calculation. 

 Mere finger-counting, mere head-reckoning, even with the 

 aid of a regular decimal notation, could not have sufficed 

 for numbering the days in a year ; much less the years, 

 months, and days between eclipses. Consequently there 

 must have been a mode of registering numbers ; probably 

 even a system of numerals. The earliest numerical rec- 

 ords, if we may judge by the practices of the less civilized 

 races now existing, were probably kept by notches cut on 

 sticks, or strokes marked on walls ; much as public-house 

 scores are kept now. And there seems reason to believe 

 that the first numerals used were simply groups of straight 

 strokes, as some of the still-extant Roman ones are ; lead- 

 ing us to suspect that these groups of strokes were used to 

 represent groups of fingers, as the groups of fingers had 

 been used to represent groups of objects — a supposition 

 quite in conformity with the aboriginal system of picture 

 writing and its subsequent modifications. Be this so or 

 not, however, it is manifest that before the Chaldeans dis- 

 covered their Saros, there must have been both a set of 

 written symbols serving for an extensive numeration, and 

 a familiarity with the simpler rules of arithmetic. 



Not only must abstract mathematics have made some 

 progress, but concrete mathematics also. It is scarcely 

 possible that the buildings belonging to this era should 

 have been laid out and erected without any knowledge of 

 geometry. At any rate, there must have existed that ele- 

 mentary geometry Avhich deals with direct measurement — 

 with the apposition of lines ; and it seems that only after 

 the discovery of those simple proceedings, by which right 

 angles are drawn, and relative positions fixed, could so reg- 

 alar an architecture be execut(id. In the case of the other 

 division of concrete mathematics — mechanics, we have defi- 



