 Descartes solved the problem of dimensionality by showing how arithmetical operations performed on lines never transcend the line. For example, if line AB is the unit (see  Fig. 8),  and I want to multiply line BD by BC, “I have only to join the points A and C, then to draw DE parallel CA, and BE is the product of this multiplication” (AT 6: 370, MOGM: 177–178). Descartes’ procedure is modeled on similar triangles (two or more triangles whose sides may have different lengths but whose angles are equal). The sides of all similar triangles are proportional to one another (e.g., triangle ACB is similar to triangle DEB, such that BC is proportional to BE and BA is proportional to BD, etc.) (see Euclid’s  Elements VI.4–5  [1908: [2] 200–204]). By exploiting the theory of proportions, Descartes could easily show that BA:BD=BC:BE, or \(1:a=b:c\) (e.g., \(1:2=2:4,\) so that \(2•2=4,\) etc.), in which case \(a•b=c\) or \(\textrm{BD}•\textrm{BC}=\textrm{BE}.\) The multiplication of two or more lines never produces a square or a solid, but only another line segment that bears a definite (proportional) relation to the other line segments. All magnitudes can easily be compared to one another as lines related to one another by some measure or proportion, effectively opening the door to the unrestricted use of algebra in geometry.