St. John's

The Powers of Euclid's Book 1

01 October 2019

Class: Mathematics Tutorial — Fall 2019

In this essay, the "powers" of Book 1 of Euclid will be examined. It will be shown that the powers can be divided into two categories "constructions" and "assertions". The differences in the use of these categories of powers will then be investigated.

A power is the authority by which a step in a proposition is performed. A proposition once proven becomes a power that can be used in future propositions. The use of the powers then is a sort of barometer for the use of proven truth. If the use of a proposition as a power can be understood then the proposition itself is better understood. In an initial attempt to understand the powers I found that the powers can be put into two clearly separate categories.

The first sign of the dichotomy to be demonstrated shows itself in the postulates. The first three postulates are clearly different from the last two. The first 3 postulates grant an ability to produce or alter an object in geometric space.¹ The first three postulates are of the type that we will call constructive and the second two are of the type that we will call assertive. The assertive postulates allow statements about one object to imply true statements about another object or for statements about one property of an object to imply true statements about another property of the same object.

A similar but more subtle distinction can also be found in the common notions. The common notions on first reading are entirely assertive. However a closer reading will find that common notions 2 and 3 are at least in part constructive. Common notions 2 and 3 assert that for a given pair of objects which are equal if they are added to or subtracted from by the same amount than the resulting objects are also equal. This assertion seems to imply that it is possible to add or subtract from objects in some way. The ability to make this distinction in the common notions and postulates prompted the question "Can the propositions also be divided into constructions and assertions?".

The very first proposition in its enunciation states "On a given finite straight line to construct an equilateral triangle" Emphasis mine. From the enunciation, it is obvious that the result of this proposition is the ability to create something in geometric space. This proposition can be classified as a construction. A second worthwhile example of a proof which results in a new power of construction is proposition 10. Its enunciation reads "To bisect a given finite straight line" This enunciation does not contain the words "To construct" it is not obvious that the power granted is a constructive one. But at the end of the proof that which was given has been transformed into that which was wanted. The alteration of an object in geometric space that can be applied in any future proposition motivates the classification of proposition 10 as constructive.

The first proposition that could obviously be classified as an assertion is proposition 4

"If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides."

The if-then structure of this proposition does imply that the assertion is conditional. This is not entirely unexpected. The definitions provide some absolute assertions and if any assertion from a proposition was absolute it would rightly be a definition. The conditionality of this assertion then is not a barrier to the classification system being proposed. I do want to note however that what is asserted in proposition 4 is equality. There are other types of assertions. In proposition 7 we can see another power of assertion proven.

"Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end."

Proposition 7 is a deceptive proposition it has the word constructed in its enunciation. It is about the construction of a triangle but it is also an assertion that the triangle being imagined cannot be constructed. At the end of proposition 7 there is no alteration of an object that was not possible before. There is no new object that can be constructed that could not be constructed before proposition 7. It could be argued that there is in fact one less object that can be constructed. In either case it is clear that Proposition 7 is asserting something other than equality. For ease of reference, I am going to call that thing non-existence.

It should be clear from this sample of propositions that some propositions grant constructive powers and others grant assertive powers. This, combined with the dichotomy in the postulates and common notions, should be sufficient to justify classifying all the propositions of Book 1 as either granting a constructive or an assertive power. When the classification scheme is applied to all of Book 1 the final count on constructive vs assertive powers granted by propositions is 14 constructions and 34 assertions.

The full classification is:

Constructions: 1, 2, 3, 9, 10, 11, 12, 22, 23, 31, 42, 44, 45, 46

Assertions: 4, 5, 6, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 47, 48

When examining the full classification of Book 1, it is noticed that the propositions are in clusters. There appear to be groups of constructive powers followed by some proofs of assertive powers followed again by more constructive powers. In fact, the propositions can be listed sequentially and still have the better part of their classification be visible.

The constructions when listed sequentially are: {1,2,3}; {9,10,11,12}; {22,23}; {31}; {42}; {44,45,46}

The Assertions when listed sequentially are: {4,5,6,7,8}; {13,14,15,16,17,18,19,20,21}; {24,25,26,27,28,29,30}; {32,33,34,35,36,37,38,39,40,41}; {43}; {47,48}

Together: {1,2,3}; {4,5,6,7,8}; {9,10,11,12}; {13,14,15,16,17,18,19,20,21}; {22,23}; {24,25,26,27,28,29,30}; {31}; {32,33,34,35,36,37,38,39,40,41}; {42}; {43}; {44,45,46}; {47,48}

When viewed as a series of continuous groups the sets of propositions appear to almost punctuate one another. These groups are first attempt at separating the proposition which grant a constructive power from those which grant an assertive power. As the groups are examined they may change slightly or even be found to be a coincidence.

In each group a tally of the uses of previously proven assertive and constructive powers will be kept. This tally can be used to compare the relative incidence of the use of each type of power. The use count of the power types may yield valuable information about the relationships between the powers and the structure of the book or the methods of geometry.

The first construction being the first proposition can of course use the authority of no other proposition. However the proposition can or rather must, use the definitions, postulates and common notions to prove that the construction it asserts is possible. For the purpose of our examinations though we are interested only in the use of the postulates and common notions² because they can be classified as constructive or assertive.

In the first group of propositions (1-3) since no assertive propositions have been proven yet only constructive powers can be used in the proofs. In this group there are a total of 2 uses of previously proven propositions. For a running total of (2) uses of constructive powers and (0) citations of assertive powers.

The second group of propositions (4-8) are all constructions. The proofs in this group use 2 constructive powers and 4 assertive powers for a running total of 4 constructive power uses and 4 assertive power uses.

Propositions 9-12 are constructions. To prove the constructions 5 constructive powers are used and 5 assertive powers are used for a running total of 9 and 9.

Propositions 13 through 21 are all grant new assertive powers. Their proofs use 6 constructive powers and 12 assertive powers. For running total of 15 construction citations and 21 assertion dependencies cited.

Propositions 22 and 23 are constructions their proofs use 2 constructive powers and 2 assertive powers resulting in a running total of 17 constructions used and 23 assertions used to complete the proofs thus far.

Propositions 24-30 are all assertions and they use only 1 power of construction but use 20 powers of assertion for a running total of 18 construction powers used and 43 assertion powers used. This kind of wild imbalance warrants a pause in the tally of the use of powers to investigate the origin of the sudden massive disparity in power use.

The one construction used in the set of propositions (24-30) is proposition 23. It is used in proposition 24, which is itself used in proposition 25. Propositions 24 and 25 are both about triangles. Propositions 27 through 30 however are about parallel lines. With this information it may be appropriate to split the group into two. One group for propositions 24-26 which prove assertions about triangles and another for propositions 27-30 which prove assertions about parallels. While altering the group structure will not change the running total of powers used (C:18-A:43), it is worth noting that the second group(27-30) does not use any powers of construction.

Since the 27-30 group is the beginning of an exploration of parallelism, it may be of interest to contrast the propositions with the beginning of the exploration of triangles which began in the very first proof with a construction. The beginning of the exploration of parallels however begins with assertions and proceeds into a single construction about parallels in proposition 31³ before proceeding into a series of propositions about parallelograms that arguably⁴ continues until the end of Book 1.

Proposition 31 itself a construction. The proof uses one construction and one assertion in its proof. For a running total of 19 constructions and 44 assertions used in the proofs so far.

The next groups of propositions (32-41) are proofs of assertive powers. They use 4 powers of construction and 28 powers of assertion. For a running total of 20 and 72. All 4 uses of a constructive power are uses of proposition 31 which allows for the creation of a line parallel to a given line. The thematic difference that was first visible in the 24-30 group is now blatantly obvious. The complete lack of use of earlier powers of construction is a profound give away that the theme has changed.

Proposition 42 is a proof of a new constructive power and it rests on the use of 2 constructive powers and 1 assertive power. Proposition 43 proves an assertion and rests on the power of only one other proposition an assertion. The running total is now 22 uses of a constructive power and 74 uses of an assertive power. Despite the opportunity to use more constructions in this second theme the author declines to do so.

Propositions 44,45,46 all give proof of the ability to create a parallelogram of some type these proofs use 5 powers of construction and 9 powers of assertion. One of the powers of construction used is proposition 11 which is not about triangles and proposition 31 is used 3 times it seems that many of the powers from the first theme will continue to go unused.

All that remains now is to examine the powers used in propositions 47 and 48. Proposition 47 cites propositions 46, 14, 4, & 41 twice the only constructive power used is proposition 46. 48 being the converse of 47 cites only proposition 47 and proposition 8 resulting in a final tally for book one of 88 uses of a power of assertion and 29 uses of a constructive power.

While a triangle is central to proposition 47 the preceding propositions about triangles do not seem to be central to the proof. Proposition 47 is often hailed as a sort of summit of book 1 but from looking at the citation set it appears to only be a summit of everything after 26. It is of course possible that first theme has a summit which is used extensively someplace early in the second portion of Book 1 which in turn holds up 45 or 47 as a summit of book one in its entirety. However to establish this would require a much more elaborate analysis than what is performed here.

The final count being so lopsided bears some investigation. Why are powers of assertion used so much more frequently than powers of creation. It may simply be that powers of assertion are more numerous. Of the 48 propositions in book 1 14 are constructions and 34 are assertions for a ratio of ~2.4 assertive powers per constructive power but the ratio of 88 uses of assertive powers and 29 uses of constructive powers is ~3.03 if the difference in uses of the powers was due to their differing amounts we would expect the ratios to be the same. Since the ratios are not the same there must be some other reason for the disparity. It may have to do with the portion of the book that is related to parallelism and that parallelism rests on only a single construction. Parallelism seems to be different from triangles or parallelograms in that a parallelism is relatively simple to construct but is complex to investigate and make assertions about.

It is also worth examining the distribution of the powers. Of the 14 constructive powers 9 occur before 26 and only 5 after that is nearly half as many in the second theme as in the first portion of the book. Part of this can be attributed to parallelism as examined before but it seems that there are constructions which could be proven which are for some reason ignored. It could be that Euclid sees the constructions as sort of incidental to the assertions, as a sort of scaffolding for the assertions. Proposition 46 certainly appears to function as a sort of scaffolding for proposition 47. The ability to produce a square is not used elsewhere in book one and proposition 46 could have appeared right after proposition 34 since it uses no power defined after proposition 34. The wait to define 46 until it is needed in proposition 47 is sufficient justification for treating at least this one construction as a sort of scaffolding.

A power of construction would seem to be so much more useful to bring into existence an object in geometric space but perhaps the assertions are more powerful. The ability to provide with 100% certainty a truth about a property of an object in geometric space. To assert truth with so much certainty is the sort of power only an oracle or a god could have. Carpenters and stone masons build houses of certain dimensions with some regularity but they cannot declare an absolute truth. The great number of assertions relative to constructions serves as evidence that Euclid must have found them much more useful. Alternatively it may be that because a construction is so useful that less can be done with more and their lack of use actually demonstrates that the constructive power is more useful than the assertive which must be used over and over again to accomplish the proofs.

Whatever the reason for the disparity in the number of and use of constructive powers vs assertive powers it is clear at least that the disparity exists. It is clear that the disparity is greater in the second portion of the book. The change in the disparity provides evidence that the book has two themes. The use of the powers provides evidence but does not prove either way that proposition 47 may not be a summit or end goal of book 1.

There are of course some questions that fall out of the analysis that have not been examined. It is clear from arithmetic, that some proofs must make no use of a power of construction. An analysis of the proofs which use no constructive power may provide information on how Euclid views the difference between the powers. We may also ask are all constructions the same? It was found that some assertions are about equality and some are about other properties such as existence or parallelism. Constructions obviously could not all be for the same object or else they would be redundant but there may be some other way of grouping constructions such as separating the true constructions from those that merely alter an object. Investigating these questions may also provide an answer to the still open question of why the constructive powers are so limited in their declaration and use compared to the assertive powers.

The conclusions stated now may feel like paltry gains and may stand as an indictment of the analytical method of investigation that has been used to make these conclusions. To these charges I make no defense.