Geometry

# Analysis and Synthesis in Geometry

Pappus of Alexandria, Mathematical Collection VII. 1–3.2 Translation of T. L. Heath History of Greek Mathematics II. 400–401

The so-called

[Treasury of Analysis] is, to put it shortly, a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving [the construction of] lines, and it is useful for this alone. It is the work of three men, Euclid, the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis.

*Analysis*, then, takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we admit that which is sought as if it were already done (

) and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (

).

But in *synthesis*, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what before were antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.

Now analysis is of two kinds, the one directed to searching for the truth and called *theoretical*, the other to finding what we are told to find and called *problematical*. (1) In the *theoretical* kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (*a*) if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (*b*) if we come upon something admittedly false, that which is sought will also be false. (2) In the *problematical* kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (*a*) what is admitted is possible and obtainable, that is, what mathematicians call *given*, what was originally proposed will also be possible, and the proof will again correspond in the reverse order to the analysis, but if (*b*) we come upon something admittedly impossible, the problem will also be impossible.^{1}

<. . . So much, then, for the definition of analysis and synthesis. Of the books already mentioned the list of those forming the

is as follows:>

^{2}
Euclid’s *Data*, one Book, Apollonius’s *Cutting-off of a ratio*, two Books, *Cutting-off of an area*, two Books, *Determinate Section*, two Books, *Contacts*, two Books, Euclid’s *Porisms*, three Books, Apollonius’s *Inclinations* or *Vergings* (

), two Books, the same author’s

*Plane Loci*, two Books, and

*Conics*, eight Books, Aristaeus’s

*Solid Loci*, five Books, Euclid’s

*Surface-Loci*, two Books, Eratosthenes’s

*On Means*, two Books. There are in all thirty-three Books, the contents of which up to the

*Conics* of Apollonius I have set out for your consideration, including not only the number of the propositions, the

*diorismi*^{1} and the cases dealt with in each Book, but also the lemmas which are required; indeed I have not, to the best of my belief, omitted any question arising in the study of the Books in question.

^{2}
# AN EXAMPLE OF ANALYSIS AND SYNTHESIS

Pappus, Mathematical Collection VII, Proposition 108 (p. 836.24 [Hultsch])^{3}

Given two points *D*, *E* within a given circle. To draw lines from *D* and *E* meeting the circumference at *A*, so that if these lines are produced to meet the circumference at *B* and *C*, *BC* will be parallel to *DE*.

ANALYSIS: Suppose the construction done. Draw *BF* tangent to the circle at *B*. Then

*B*,

*F*,

*A*, and

*E* lie on the same circle.

rectangle

*BD.DA* = rectangle

*FD.DE*.

But rectangle *BD.DA* is given [Euclid, *Data* 92] (for *ADB* is drawn from a given point *D* to a circle given in position).

*FD.DE* is given.

But *DE* is given.

*FD* is also given.

And *D* is given.

*F* is also given.

Now *FB* has been drawn tangent from a given point *F* to a circle given in position.

*FB* is given in position.

Now the circle, too, is given in position.

point

*B* is given.

Again, point *D* is given.

*BD* is given in position.

And, since the circle is given in position, point *A* is given.

But both *D* and *E* are given.

lines

*DA* and

*AE* are given in position.

SYNTHESIS: The synthesis of our problem is as follows:

A. *Construction:*

Let *ABC* be the circle given in position and *D*, *E* the two given points. Let any line *ADB* be drawn.^{1}

Make the rectangle *ED.DF* equal to the rectangle *AD.DB*.^{2}

From *F* draw *BF* tangent to the circle.

Draw *CEA*.

B. *Proof*:

Since

(for points

*A*,

*F*,

*B*, and

*E* lie on the same circle), and also

(for

*FB* is a tangent and

*BA* a secant),

and *BC* is parallel to *DE*. Q.E.D.

^{1} This statement could hardly be improved upon except that it ought to be added that each step in the chain of inference in the analysis must be *unconditionally convertible*; that is, when in the analysis we say that, if *A* is true, *B* is true, we must be sure that each statement is a necessary consequence of the other, so that the truth of *A* equally follows from the truth of *B*. This, however, is almost implied by Pappus when he says that we inquire, not what it is (namely, *B*) which follows from *A*, but what it is (*B*) from which *A* follows, and so on.

[For an example of analysis and synthesis see the following selection. Edd.]

^{2} The material within < > is inserted to complete the translation. [Edd.]

^{1} "The *diorismos* is a statement in advance as to when, how, and in how many ways the problem will be capable of solution" according to a gloss on this passage, which has intruded itself into the text after the definitions of analysis and synthesis. See p. 36. [Edd.]

^{2} Aristaeus preceded Euclid in the fourth century B.C. His work here referred to is a treatise on conic sections, which were called "solid loci" to distinguish them from those that could be constructed with straight edge and compass, the so-called "plane loci." Apullonius of Perga, who followed Archimedes, toward the end of the third century B.C., brought the subject of conics to its highest development. Besides his *Conica* his treatises, including those referred to here, recorded extensions and discoveries of great importance in the geometry of line and circle as well as in conic sections. His researches in mathematical astronomy included the theory of epicycles and eccentric circles in connection with the planetary system (see p. 128). [Edd.]

^{1} The meaning is that *any* chord through *D* will enable us to obtain the product *AD.DB*. The precise location of *A* is determined only after *F* and *B* have been found.

^{2} I.e., construct *DF* a fourth proportional to *ED*, *AD*, and *DB*.