[b][u]Propositions 18 and 19[/u][/b][i][b]I.      If OA be the initial line, A the end of the first turn of the spiral, and if the tangent to the spiral at A be drawn, the straight line OB drawn from ) perpendicular to OA will meet the said tangent in some point B, and OB will be equal to the circumference of the "first circle".[/b][/i] [url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral.ps][img width=400,height=161]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral.jpg[/img][/url][b][br]Proof: [/b]Suppose OB is greater than c, the circumference of the first circle.[url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral2.ps][img width=400,height=184]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral2.jpg[/img][br][br][/url]Suppose OB is less than c, the circumference of the first circle.[url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral3.ps][img width=400,height=255]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral3.jpg[/img][br][/url]Since OB is not less than or greater than c, OB = c.[url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral4.ps][img width=400,height=215]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/spiral4.jpg[/img][br][br][/url][b][i]II.     If A' be the end of the second turn, the perpendicular OB will meet the tangent at A' in some point B', and OB' will be equal to 2(circumference of the "second circle").[br][/i][/b][b]Proof: [/b]Suppose OB is greater than 2c', twice the circumference of the second circle.[url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/2spiral.ps][img width=400,height=274]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/2spiral.jpg[/img][br][/url]Suppose OB is less than 2c', twice the circumference of the second circle.[url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/2spiral2.ps][img width=400,height=275]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/2spiral2.jpg[/img][br][/url]Since OB is not less than or greater than 2c', OB = 2c'.[url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/2spiral3.ps][img width=400,height=278]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/2spiral3.jpg[/img][br][/url][i][b]III.     Generally, if A[sub]n[/sub] be the end of the nth turn, and OB meet the tangent at A[sub]n[/sub] in B[sub]n[/sub], then[br] OB[sub]n[/sub] = nc[sub]n[/sub],[br][/b][/i][i][b]where c[sub]n[/sub] is the circumference of the "nth circle"[br][br][/b][/i]The proof for this is similar to the proofs for the first and second circles.[url=https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/3spiral.ps][img width=400,height=274]https://personal.math.ubc.ca/~cass/courses/m309-01a/darien/3spiral.jpg[/img][/url]