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Book I  Problems the Construction of Which Requires Only Straight Lines and Circles  
How the calculations of arithmetic are related to the operations of geometry  297  
How multiplication, division, and the extraction of square root are performed geometrically  293  
How we use arithmetic symbols in geometry  299  
How we use equations in solving problems  300  
Plane problems and their solution  302  
Example from Pappus  304  
Solution of the problem of Pappus  307  
How we should choose the terms in arriving at the equation in this case  310  
How we find that this problem is plane when not more than five lines are given  313  
Book II  On the Nature of Curved Lines  
What curved lines are admitted in geometry  315  
The method of distinguishing all curved lines of certain classes, and of knowing the fatios connecting their points on certain straight lines  319  
There follows the explanation of the problem of Pappus mentioned in the preceding book  323  
Solution of this problem for the case of only three or four lines  324  
Demonstration of this solution  332  
Plane and solid loci and the method of finding them  334  
The first and simplest of all the curves needed in solving the ancient problem for the case of five lines  335  
Geometric curves that can be described by finding a number of their points  340  
Those which can be described with a string  340  
To find the properties of curves it is necessary to know the relation of their points to points on certain straight lines, and the method of drawing other lines which cut them in all these points at right angles  341  
General method for finding straight lines which cut given curves and make right angles with them  342  
Example of this operation in the case of an ellipse and of a parabola of the second class  343  
Another example in the case of an oval of the second class  344  
Example of the construction of this problem in the case of the conchoid  351  
Explanation of four new classes of ovals which enter into optics  352  
The properties of these ovals relating to reflection and refraction  357  
Demonstration of these properties  360  
How it is possible to make a lens as convex or concave as we wish, in one of its surfaces, which shall cause to converge in a given point all the rays which proceed from another given point  363  
How it is possible to make a lens which operates like the preceding and such that the convexity of one of its surfaces shall have a given ratio to the convexity or concavity of the other  366  
How it is possible to apply what has been said here concerning curved lines described on a plane surface to those which are described in a space of three dimensions, or on a curved surface  368  
Book III  On the Construction of Solid or Supersolid Problems  
On those curves which can be used in the construction of every problem  369  
Example relating to the finding of several mean proportionals  370  
On the nature of equations  371  
How many roots each equation can have  372  
What are false roots  372  
How it is possible to lower the degree of an equation when one of the roots is known  372  
How to determine if any given quantity is a root  373  
How many true roots an equation may have  373  
How the false roots may become true, and the true roots false  373  
How to increase or decrease the roots of an equation  374  
That by increasing the true roots we decrease the false ones, and vice versa  375  
How to remove the second term of an equation  376  
How to make the false roots true without making the true ones false  377  
How to fill all the places of an equation  378  
How to multiply or divide the roots of an equation  379  
How to eliminate the fractions in an equation  379  
How to make the known quantity of any term of an equation equal to any given quantity  380  
That both the true and the false roots may be real or imaginary  380  
The reduction of cubic equations when the problem is plane  380  
The method of dividing an equation by a binomial which contains a root  381  
Problems which are solid when the equation is cubic  383  
The reduction of equations of the fourth degree when the problem is plane, Solid problems  383  
Example showing the use of these reductions  387  
General rule for reducing equations above the fourth degree  389  
General method for constructing all solid problems which reduce to an equation of the third or the fourth degree  389  
The finding of two mean proportionals  395  
The trisection of an angle  396  
That all solid problems can be reduced to these two constructions  397  
The method of expressing all the roots of cubic equations and hence of all equations extending to the fourth degree  400  
Why solid problems cannot be constructed without conic sections, nor those problems which are more complex without other lines that are also more complex  401  
General method for constructing all problems which require equations of degree not higher than the sixth  402  
The finding of four mean proportionals  411 
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