396 PROCEEDINGS OF THE AMERICAN ACADEMY. 



that AB' falls along AB, and AC along AC. Assume that AC and 

 AC are commensurable. Apply the common measure to the side 

 AC, and through the points of division draw lines parallel to BC 

 and to AB. In the small triangles thus formed the parallel sides will 

 be equal by IV, and therefore the intervals cut off on AB must be 

 equal by II. In case of incommensurability the method of limits 

 may be applied.^ The case in which the two triangles fall on opposite 

 sides of the common vertex may be treated in a similar manner by the 

 aid of IV. 



8. For our future needs, the conception and the measure of area 

 are fundamental, and it is important to show that this subject may be 

 satisfactorily treated with the aid of the parallel-transformation 

 (that is, the translation) alone. Indeed, any arbitrarily chosen unit 

 intervals along any selected pair of intersecting lines determine a 

 parallelogram which may be taken as having a imit area. By ruling 

 the parallelogram into equal parallelograms by lines parallel to its 

 sides, an arbitrarily small element of area may be obtained. The area 

 enclosed by any curve may be divided into like elements by similar 

 rulings, and thus by the method of limits the enclosed area may be 

 compared with the assumed unit area.^ In particular some simple 

 propositions on areas will now be deduced. 



VI. Any parallelogram with sides parallel to those of the unit 

 parallelogram has an area equal to the product of the intervals along 

 two intersecting sides. 



8 It may be observed at this point that if two intersecting lines be taken as 

 axes of reference, if systems of measurement (as yet necessarily independent) 

 be set up along the two lines with the point of intersection as common origin, 

 and if to each point P of the plane are assigned coordinates (x, y) equal to the 

 intercepts cut off from the axes by lines through P parallel to the axes, then 

 straight lines are represented by linear equations, and conversely. For the 

 deduction of the equation of a line depends merely upon the properties of 

 triangles similar in our sense. The transformation from any such set of axis 

 to any other such set will clearly be linear. 



9 If axes be introduced as above, the area of a triangle and the area of any 

 closed curve are expressed analytically by the usual formulas. 



and I fdxdy = Lxdy = —njdx, 



in terms of our assumed unit parallelogram. The theorems on areas could 

 then be proved analytically, but the elementary geometric demonstrations 

 seem preferable. It is important to observe further that in a transformation 

 to new axes, such that 



.r = „.r' + by' + c, y = a'x' + h'y' + c', 



