Consider a series of squares, concentric and similarly situated. Let the lengths of the sides of the successive squares, stated in order of diminishing size, be Then each square extends over all the subsequent squares of the set. Also let namely, let tend to zero as increases indefinitely. Then the set forms an abstractive class. Again, consider a series of rectangles, concentric and similarly situated. Let the lengths of the sides of the successive rectangles, stated in order of diminishing size, be (), (),...(),.... Fig. 6. Thus one pair of opposite sides is of the same length throughout the whole series. Then each rectangle extends over all the subsequent rectangles. Let , tend to zero as increases indefinitely. Then the set forms an abstractive class. Evidently the set of squares converges to a point, and the set of rectangles to a straight line. Similarly, using three dimensions and volumes, we can thus diagrammatically find abstractive classes which converge to areas. If we suppose the centre of the set of squares to be the same as that of the set of rectangles, and place the squares so that their sides are parallel to the sides of the rectangles, then the set of rectangles covers the set of squares, but the set of squares does not cover the set of rectangles. Again, consider a set of concentric circles with their common centre at the centre of the squares, and let each circle be inscribed in one of the squares, and let each square have one of the circles inscribed in it. Then the circles form an abstractive class converging to their common centre. The set of squares covers the set of circles and the set of circles covers the set of squares. Accordingly the two sets are -equal.] 31. Primes and Antiprimes. 31.1 An abstractive class is called 'prime in respect to the formative condition ' [whatever condition '' may be] when (i) it satisfies the condition , and (ii) it is covered by every other abstractive class satisfying the same condition . For brevity an abstractive class which is prime in respect to a formative condition is called a -prime. Evidently two -primes, with the same formative condition in the two cases, are -equal. 31.2 An abstractive class is called 'antiprime in respect to the formative condition ' [whatever condition '' may be] when (i) it satisfies the condition , and (ii) it covers every other abstractive class satisfying the same condition . For brevity an abstractive class which is antiprime in respect to a formative condition is called a -antiprime. Evidently two -antiprimes, with the same formative condition in the two cases, are -equal. 31.3 Let be any assigned formative condition, let be the condition of 'being a -prime,' and let be the condition of 'being a -antiprime.' Thus an abstractive class, which satisfies the condition , (i) satisfies the condition , and (ii) is covered by every other abstractive class satisfying the same condition .