CONVERTED LIBRARY 1 UNIVERSITY OF CALIFORNIA. Received '^/J'Zt^'ly . , / 8c)6 . Accessions hloP / ^^^O. Class No. A \yj OUR NOTIONS OF Number and Space HERBERT NICHOLS, Ph.D. LATE IXSTRCCTOR IN PSYCHOLOGV, HARVARD IXIVERSITY ASSISTED BT WILLIAM E. PARSONS, A.B. 7B^ •ITY)j irh^ BOSTON, U.S.A. GIXN & COMPANY, PUBLISHERS 1894 JV 5 Copyright, 1894 BY HERBEKT NICHOLS ALL RIGHTS RESERVED CONTENTS. Page. Introductiox ......... V ExPERi.MEXT A. — With Pins set in a Straight Line . . 1 Experiment B. —With Pins set in Triangles and Squares 21 Experiment C. — With Lineal Figures .... 40 ExPERi.MENT D. — With S(jlid Figures . . .43 Experiment E. — With Moving Pencil .... 57 Experiment F. — Comparing Horizontal ami Vertical Dis- tances 63 A Study of the Kesults ....... 71 Number .......... 71 Distance 86 Number-Judgments based on Two Dimensions . . Ill Distance-Judgments based on Two Dimensions . . 126 Judgments of Figure 142 The ilass, Intensity and Time Elements of Distance- Judgments ........ 147 ExPERi-MENT G. — With a Single Pin ..... 156 Experiment H. — Education of Artificial Space-Relations. 172 General Survey and Summary ...... 177 INTRODUCTION. My thesis I briefly state as follows : Our brain habits, with the modes of thought and of judgment dependent thereon, are morphological resultants of definite past experiences : our experiences, and those of our ancestors. Each limited experience does its share toward fixing a limited habit. The experiences most common to our various regions of skin, differ widely one from another ; those of the tongue, from those of the fingers ; those of the fingers, from those of the abdomen, and so on. Our habits of judgment, based on these several avenues of experience, ought therefore, when compared with each other, to betray permanent characteristics running parallel with the local differ- ences of anatomy, of function, and of experience, which give rise to them, and in which they are rooted. Investigation proves this to be the case. It shows that our judgments of the same outer facts, such as of number and of distance, vary greatly when mediated by different tactual regions. And what is of greater importance to the science of psychology, these varia- tions in judL;ment bear distinguishing ear-marks of the VI INTRODUCTION. kinds of experience out of which, and by reason of which through life, they have slowly risen. It is our purpose to study these. Through compari- son of the different constants and variables in certain judgments, which we shall subject to experimental proof and analysis, we aim to discover somewhat regarding the fundamental laws governing the past genesis and the present formation of our judgments, and of the movements of mental processes in general. As the last words of this Introduction, I wish to thank Professor Miinsterberg for permitting one of his students to assist me with this research during an entire year. And with deep appreciation, and pleasant recollections, I record the patient labor and able service which Mr. Parsons has continually contributed to the work. OUR NOTIONS OF NUMBER AND SPACE. EXPERIMENTS A, B, C, D, E, F, G, and H. These several experiments form a set. "We shall first present the method, and the bare results of each one separately, then study them collectively. EXPERIMENT A. WITH PINS SET IX A STRAIGHT LIKE. Apparatus. — Heavy cardboard was cut in strips 7 or 8 mm. narrower than the pins to be used. The pins were the familiar household article ; they were run through the whole width of the strip, which held them firmly, their ends projecting like the teeth of a comb. Thirty-six cards were thus prepared, or 9 sets of 4 cards each. The '' 9 sets " corresponded to the 9 distances experimented with ; and by " distance " we shall always denote the distance between the end pins of tlie line of pins. The 9 distances embraced the even and the half cmm. from 1 to 5 inclusive. The 4 cards of each " distance set " were fitted with 2, 3, 4, and 5 pins respectively. These pins (Avhen 2 OUK NOTIONS OF NUMBER AND SPACE. more than 2) were spaced equally apart for each card ; these sub-distances, of course, varying on each separate card. Thus prepared, our 36 cards represented 9 categories of "distance,"' and 4 categories of "number" for each distance. (For the abdomen an extra 6-pin card was used.) A holder was provided for these cards, in order that the subject, wlien taking them in his hand, should learn nothing about them through the pinch of his fingers. This holder was a folded strip of sheet-steel, — the cards were dropped into its groove, with the heads of the pins resting against the metal, and the sides of the holder were then pinched together. Method. — In this experiment the subject applied the pins to himself, the cards being drawn, put into the holder, and handed to him by some one else. In applying the cards the subject was permitted to 'rock the pins back and forth on his skin. The line of direction, in which the line of pins were applied, was for each locality always the same. This was at right angles to the median line on the tongue, the forehead, and the abdomen, and longitudinally on the forearm. At first an instrument was used to regulate the pressure with which the pins were applied ; but it was soon found, the pins being sharp, that the subject's own feeling, adapting itself very sensitively to the conditions OUR NOTIONS OF NUMBER AND SPACE. 3 of best judgment under varying conditions of thickness and toughness of skin, was a better " control " for the " constancy of pressure " than any mechanical contriv- ance could possibly be. Proper care was used to avoid complications due to fatigue or to changes of temperature. Exjilanatlons of the Tables. — In this experiment : A four persons were experimented upon as follows. B, a student of biology ; L, a student of psychology ; P, Mr. Parsons ; and N, myself. Each card of pins was applied 100 times to each person. The " distance " categories are indicated in the left-hand vertical column and govern the horizontal line of figures opposite to them, across the page. The "number" categories, show- ing the number of pins in each card, are indicated by Roman numerals in the top horizontal heading, and govern the vertical column of figures below, to the bottom of the page. The main body of figures shows averages calculated from 100 applications of each card to each of the four persons, i.e., from a total of 400 applications. The four main horizontal divisions of the tables show as follow : — The First : — Shows the number of times, per hun- dred times applied, that the number of pins Avas judged correctly. The Second : — ShoAvs the per cent, error made in judging the number of pins. 4 OUR NOTIONS OF NUMBER AND SPACE. The Third : — Shows the number of times per hundred times applied, that the distance was judged correctly. The Fourth : — Shows the per cent, error made in judging the distance. Tables 1, 2, 3, and 4 are wholly '' regular " according to the foregoing explanations. Table 5. — The question arose as to what part the rocking of the pins back and forth on the skin, which was permitted the subject, played in making his judg- ment. Or to put the matter more psychologically, the question rose as to how far such judgments were direct, and how far complex and reasoned out. To throw light on tliis matter, a set of tests was made upon the fore- arm, which differed from the regular experiments in that the pins were only permitted to be pressed upon the skin steadily and evenly throughout the whole line, and but three times in regular succession, at intervals one second apart. The results so obtained are reported in Table 5. Table 6. — Particularly it was suspected, that our judgments of the number of pins in a given card were reasoned out somewhat as follows : that, feeling perhaps, the two end pins widely apart, and two inter- mediate pins nearer together, or even one pin at some intermediate point, we then said : — " since the inter- mediate pins are spaced equally there ought to be so many pins"; thus arriving at the final estimate by OUR NOTIONS OF NUMBER AND SPACE. 5 mathematical calculations based on the partial data actually given in the impression. It was found that, Avith practice, this sort of reckoning process could be largely suppressed by volition ; by concentrating our attention upon the number of points felt, and strenuously shutting out all else from the impression. If we could not suc- ceed in this perfectly, it was well to compare results obtained by this method, with those where, as in the regular experiments, every possible aid was given to forming the judgments. Table G presents such results. Table 7. — It being a main proposition of this re- search to compare results obtained upon dissimilar regions of the body, it was requisite that all the results should be obtained under conditions as similar as possible. As a matter of fact the different regions were for each separate experiment worked upon suc- cessively, and the full set of tests was finished for one region before proceeding to another. At the end, the question arose: how were the later results influenced by the considerable amount of skill and practice acquired in the foregoing work? To test this, still another series, perfectly regular in its method, was taken upon the forearm, at the very end of all our work. Table 7 contains its results. The regions were first Avorked on in the following jo^^err^v^-xtongue, forehead, fore- [uhivee:it7) 6 OUR NOTIONS OF NUMBER AND SPACE. arm, and abdomen, and the corresponding tables are arranged in similar order. Tables 8 and 9. — For theoretical reasons to be reached in our future discussions, it became desirable to have, for comparison with our other results, a set of judgments from impressions where the distance cate- gories remained the same as in our regular experiments, but where the number of pins, or points, in each line should be increased to a maximum, or to infinity. That is, wherein a straight line, or straight edge should be pressed upon the skin, instead of pins set at intervals in a line. Accordingly a series of tests was made upon the forearm, with a set of cards, cut to proper lengths from thin, hard card-board, the whole length of the edge of the card being pressed directly upon the skin. Table 8 shows results obtained Avith them, according to the regular method of permitting the subject to ''rock'' the card upon the skin. Table 9 shows results comparative with those of Table 5, where the cards were pressed evenly and steadily three times in suc- cession, at intervals one second apart. Table 10 is a general summary of the foregoing tables, and aids in comparing the different regions worked upon. Note. — A glance at any of our tables shows them divided into sub-tables, or blocks. Each block bears a number, in parenthesis, in its upper middle portion. I shall always refer to these minor OUR NOTIONS OF NUMBER AND SPACE. 7 tables as " blocks," identifying them by their proper numbers. The total number of blocks is 356. But, owing to the great expense that would be incurred, I am unable to publish the results obtained from the individual subjects, and can lay before the present reader only the blocks and figures which present the averages calculated from the four subjects. The original figures, however, are at the service of any one who cares for them. OUR NOTIONS OF NUMBER AND SPACE. Table 1. Bzperiment A. — Pins in straight line. TONGUE. Subject Distance between End Pins (Centi- meters) Average of N. P. L. ajtd B. Xo. PiXS IX LiXE 11 III IV V Averages o 5 (5) g 1 2| 1 100 97 96 99 Ph 1.5 100 99 98 99 o 'm o S- 2 100 100 97 98 a e ^ " 2.5 100 100 98 97 a ii 3 100 99 97 96 Averages 100 99 97.3 97.8 98.5 '4a O a 5 (10) ir. w d . 1 + 1.1 + 1.2 - .3 - ^ "^ 1.5 + .3 + .5 - .2 S «- ° s •> + .3 - .4 § (^2-^ 2.5 + .5 - .2 o 3 + .3 - .2 - .8 Averages + .3 + .5 - .4 + .075 (15) 'A III 1 1.5 98 08 99 91 97 91 100 97 98.5 94.2 »s « s3 a. 2 95 94 95 97 95.2 f- IS^ 2.5 91 95 96 98 95.0 n «i 3 98 94 93 97 95.5 O Averages 96.0 94.6 94.4 97.8 95.68 (20) -AA .£ 1 + 1.0 + .5 + 1.5 + -7 Q go's 1.5 + .7 + 3.0 + 2.5 + 1.3 + 1.9 D " ^-f 2 + 1.2 + 1.2 + .1 - .4 + .5 (So-^ 2.5 + 1.4 + .4 + .6 + .2 + .6 3 - .3 -1.0 -1.2 - .5 - .7 Averages + .8 + .8 1 + .7 + .1 1*=^ 10 OUR NOTIONS OF NUMBER AND SPACE. Table 2. Experiment A. — Pins in straight line. FOREHEAD. Subject Distance between Average of N. P. L. AND B. End Pins No Pins in Line (Centi- meters) II Ill IV V Averages !i (25) IZ •g-s 1 5 16 61 64 Ph |§l 1.5 28 27 76 41 o 2 2.5 58 !K5 8 32 60 57 30 26 a ^2 3 99 60 52 26 g u Averages 57.2 28.6 61.2 37.2 46.0 Iz; ISi O (30) w 6 . 1 + 84.2 + 43.5 + 1.5 - 8.7 2; gi^'g 1.5 + 58.8 + 24.8 -9.0 -14.9 0) t. fl CM 2 ^ 2 2.5 + 30.5 + 1.5 + 25.1 + 22.8 -8.3 -2.0 -17.8 -20.0 O 3 + .2 + 4.3 -8.0 -22.0 Averages + 35.0 + 24.0 -5.2 -16.7 + 9.275 !i (35) u 1 2 1 1 1.5 30 57 65 30 78 43 90 36 , 65.7 43.7 !5 '" al 2 51 47 45 40 45.7 2 ^ & 2.5 68 49 51 48 54.0 3 82 78 60 56 69.0 Averages 57.6 55.1 55.4 54.0 55.62 m H is (40) S 1 + 42.1 + 16.4 + 11.2 + 7.0 + 19.2 g«^ OJ t. S Ph 2 -^ 1.5 2 + 4.7 + 1.0 + 6.7 + 1.1 + 1.2 - 9.3 - 9.9 -11.2 + .7 - 4.6 •-S 2.5 + 3.6 - 6.8 - 7.3 — 12.3 - 5.7 o 3 - 3.2 - 6.1 -11.1 -14.1 - 8.6 Averages + 9.6 + 2.5 - 2.5 - 8.1 + 2.0 OUR NOTIONS OF NUMBER AND SrACE. 11 Table 3. Experiment A. — Pins in straight line. FOREARM. Sdbject Distance between Average of X. P. L. and B. End Pins No . PrNS IN Line (Centi- meters) II III IV V Averages M (45) 2 0) w 1 26 31 45 23 P^ ■§ 2 .s 1.5 39 34 37 19 o « S s 2 54 37 31 20 « 2 "- rt 2.5 59 39 41 13 ea P 3 64 27 30 14 Averages 48.4 33.6 36.8 17.8 34.2 'A O s (50) — o . 1 + 70.0 + 18.1 -11.7 -27.3 »S g£| 1.5 +47.0 + 9.3 -14.2 —27.7 S 2 + 43.0 + 7.1 -15.1 -23.9 o Q PM g-^ 2.5 + 38.0 + .3 -18.3 —29.9 3 + 20.0 + .9 -21.1 -29.1 Averages + 45.4 + 7.2 -16.1 -27.6 + 2.225 8| (55) 4> ■" M Q —■ "3 1 » 1 20 35 42 46 35.7 m 1.5 47 33 29 27 44.0 S5 H 2 2.5 55 41 50 41 44 39 40 40 47.2 40.2 3 57 40 49 48 48.5 Averages 44.0 39.8 40.4 40.2 43.16 en H 3 (60) ? ■s" '^ ^ 1 + 69.1 + 56.0 +44.3 + 40.0 + 52.4 0> g^^ 1.5 + 33.1 +29.4 +25.1 + 21.0 + 27.1 L3 2 2.5 - .2 - 4.3 + .5 - 1.6 - 6.9 - 7.3 + .9 - 9.0 - .1 - 6.9 3 - 9.1 -12.3 -14.9 -15.1 -12.8 Averages + 17.7 + 13.3 + 9.1 + 7.6 + 11.9 12 OUR NOTIONS OF NUMBER AND SPACE. Table 4a. Experiment A. — Pins in straight line. ABDOMEN. Subject Distance between End Pins Averages of N. P. L. AJST) B. No. Pins in- line (Centi- meters) II III IV V VI Averages 8 03 (65) ft ^ O'-i 1 7 32 59 55 £.^ 1.5 8 21 57 52 oft o ft 2 11 16 55 47 ^ ^ 2.5 10 20 51 42 5 3 29 28 44 44 ^ II 3.5 34 29 44 37 h m 4 52 30 49 35 o s 4.5 08 36 41 26 d 5 79 38 36 28 14 a Averages 34.1 27.8 48.4 40.7 14.0 33.0 ^ > (u o H (70) 6 1 + 109.0 + 34.1 + 12.1 -13.3 O 1.5 + 09.4 + 30.8 + 12.4 -13.9 1-5 2 2.5 + 00.1 + 78.8 + 38.7 + 37.2 + 10.1 + 9.1 -14.9 -16.2 3 + 62.3 + 20.1 + 9.1 -J7-4 o 3.5 + 53.0 + 30.1 + 6.3 -18.2 a 4 + 42.4 + 25.1 + 4.2 -21.4 4.5 + 31.2 + 11.4 - 6.1 -24.9 Ph 5 + 21.2 + .2 -12.1 -25.5 -22.2 Averages + 6 + 27.3 + 5.2 -18.4 -22.2 + 11.44 OUR NOTIONS OF NUMBER AND SPACE. 13 Table 4b. Experiment A. Pins in straight line. ABDOMEN. Subject Distance between Averages of N. P. L . AND B End Pins No. Pins in Line (Centi- meters) II III IV \' VI Averages 1 (75). >^ a-A 1 42 53 66 78 59.7 £l 1.5 25 24 31 30 27.5 11 2 28 20 28 24 27.2 ■^l 2.5 28 28 26 25 26.7 Soi 3 28 27 27 25 26.7 a 1^-5 3.5 23 21 18 20 20.5 1 4 27 25 22 23 24.2 < 4.5 2() 26 24 20 24.0 tf. 5 48 44 40 39 38 40.2 '"' Averages 30.6 30.8 31.3 31.6 38.0 30.74 C T. H ie » 5/ 1 + 37.0 + 29.8 (80) + 21.0 + 16.1 + 26.0 i-s o 1.5 + 8.0 + 8.4 + 3.3 - 8.1 + 3.1 p 3> 2 + 12.8 + 9.9 + 1.0 - 1.2 + 5.6 2.5 + 12.3 + 1.1 - 3.1 - 6.2 + 1.0 iM 3 3 + 13.2 + 9.2 + .5 - .3 + 5.6 C 3.5 + 12.0 + 8.2 + .6 - .4 + 5.1 C 4 + 8.9 + .4 - .4 - .4 + 2.1 v 4.5 - .8 - .5 - 4.2 - 7.7 - 3.3 I 5 -11.7 -12.5 -14.7 — 15.5 -16.2 -14.1 Averages + 10.3 + 6.0 + .4 - 2.6 -16.2 + 3.46 14 OUR NOTIONS 0¥ NUMBER AND SPACE. Table 5. Experiment A. — Pins in straight line, (a) FOREARM. Pressing evenly, three times only. SlIBJECT Distance between Average OF N. AND P. End Pins No. Pins in Lute (Centi- meters) II III IV V Averages 8 S (83) "=> -s. Ph 0) *i ^ 2 £ 1 1.5 13 14 21 22 35 26 33 34 o 2 2.5 26 20 17 22 29 34 40 32 n 3 41 21 40 32 Averages 22.8 20.6 32.8 34.2 27.6 ^ m o £ (86) H *^' 6 . 1 + 78.0 + 26.0 - 5.2 -17.6 Z; H ^ c^ * 1.5 + 79.5 + 26.7 - .8 -16.4 (P (- s 2 2.5 + 75.5 + 71.5 + 28.5 + 21.2 - 4.2 - 4.1 — 14.4 -18.0 o 3 + 66.5 + 21.0 - 4.5 -16.7 Averages + 74.2 + 24.7 - 3.8 -16.6 + 19.625 8^ (89) ? 2l 1 12 6 13 15 11.5 u 1.5 28 16 16 12 18.0 Iz r Si 2 29 25 32 25 27.7 H 2.5 42 33 25 26 31.5 ^ Ti o 38 41 40 40 39.7 b Averages 29.8 24.2 25.2 23.6 25.86 H >5 o a (92) u « w . 1 + 86.5 + 104.0 + 106.0 + 98.5 + 98.7 1 £ S) 1.5 +29.5 + 41.1 + 29.8 + 55.8 + 38.8 D t. o "2 2 + 1.6 + 2.8 + 6.5 + 9.9 + 5.2 '-s P4 O ^ 2.5 - 2.2 - 2.7 1.1 - 2.5 - 2.1 w o 3 -13.4 - 13.6 — 13.5 -15.6 -14.0 Averages + 20.4 + 26.3 + 25.6 + 19.2 + 11.94 OUR NOTIONS OF NUMBER AND SPACE. 15 Table 6. Experiment A. — Pins in straight line. (b) FOREARM. Attention given only to number points felt. Subject Distance between Average of N. AND P. End Pins No. PlN.S IX Line (Centi- meters) II Ill IV V Averages I 2 (95) 1?; 1 84 21 11 P4 •3 o j; 1.5 81 25 7 1 o iP 2 2.5 79 74 27 23 14 27 5 i^ 3 79 24 27 2 S a Averages 79.4 24.0 17.2 1.4 30.5 '•^ »5 in O s s (98) jJ o . 1 + 23.1 + 12.4 -27.0 -43.6 K gi^s 1.5 + 19.8 + 13.0 -•27.2 -48.9 g ^ o -^ 2 + 14.4 + 14.2 -32.9 -47.2 Q (^ 2-^ 2.5 + 19.5 + 1G.9 -33.5 -47.0 o 3 + 14.8 + 14.7 -26.6 -53.1 Averages + 18.3 + 14.2 -29.4 -48.0 -11.225 « "ai "O' 5 1 J (101) 00 0< A , i 0.2 & 1 84 62 55 51 of ti 3 ma mes 1.5 81 54 56 58 2 79 60 60 48 4* a 5 .o » ? o 1 a - 2.5 3 74 79 63 54 42 37 63 52 ^ .a, a- Averages 79.4 58.6 50.0 52.4 IG OUR NOTIONS OF NUMBER AND SPACE. Table 7. Experiment A. — Pins in straight line, (c) FOREARM. Test of improvement through practice. Subject Distance between Average OF N. AJfD P. End Pins No . Pins in Line (Centi- meters) ]I 111 IV V Averages 8 1 (104) z £ '-5 1 12 30 50 28 Ph ■c S ^ 1.5 26 24 56 26 o ■« 11 •) 38 34 48 12 K a; 0. g- 2.5 50 28 52 12 W 1^ 3 64 36 48 14 p Averages 38.0 30.4 50.8 18.4 34.4 k( C ■s (107) o -o 1 + 84.0 + 22.6 - - 8.0 -26.0 1^1 1.5 +64.0 + 18.0 - -11.0 -19.6 S u u ^ 2 + 50.0 + 21.3 - -10.5 -23.6 C ^ -•! 2.5 +40.0 + 8.0 - - 9.0 -30.0 a o 3 + 24.0 + .3 - - 8.0 -26.4 Averages + 52.4 + 14.0 - - 9.3 -25.1 + 8.0 (110) H "§ S .2 1 1.5 30 32 24 30 30 34 28 36 29.5 33.0 9^ '« s 1 2 28 50 42 38 39.5 H s S^ 2.5 40 32 30 46 37.0 R h 1^ 3 84 86 74 68 78.0 Averages 44.0 44.4 42.0 43.2 43.4 8 (113) ^ .23 1 + 48.0 + 60.0 + 55.0 + 53.0 + 54.0 o a s 5 '^ 1.5 + 25.3 + 20.6 + 22.0 + 21.3 + 23.8 2 + 20.0 + 9.5 + 9.0 + 7.0 + 11.4 •-S f^ o •- 2.5 + 3.6 - 2.0 — 4.8 - 1.6 - 1.2 U w o 3 - 2.6 - 2.3 — 11.3 - 5.6 - 5.4 Averages + 18.9 + 19.2 + 12.2 + 14.8 + 16.5 OUR NOTIONS OF NUMBER AND SPACE. 17 Tables 8 and 9. Experiment A. — Supplement. STRAIGHT-EDGE. OP TBrR^* FOREARM. («) (0) (d) Regular. (e) Pressing evenly three times only. Distance Average Average OR Length of OF 1 OF Straight-Edge N. P. L. AND B. N. P. L. AND B. 8 s (1 4) (11(3) -o .5 1 53.0 48.7 H 1.5 50.0 39.7 f. T S 0. 2 60.0 46.5 H IS" 2.5 72.0 44.5 O H 3 83.0 55.5 Averages 63.8 47.0 (1 I.-.) (117) s *i .2 1 + 31.4 + 34.8 §- S) 1.5 + 8.9 + 10.6 3 " ® -3 2 + 2.7 + 1.3 (^2-- 2.5 - .6 - .8 3 - 7.8 - 1.5 Averages + 6.9 + 8.9 18 OUR NOTIONS OF NUMBER AND SPACE. « fa o p w a 1 ■J. A g» P^ r-" ' H r. r^ "^ « S ^ 5 o § fa w < fa < fa ;?; fa fa o s fa o t^ !?; fa E -1^ OO (M O -^ « O O O T-H 1-^ CO -^ O lO CO O lO CO G5 lO •>* CO iM -^ q q CO t^ CD -t * CO CO + .60 + .20 + 11.90 + 3.46 + 11.94 + 16.50 d 00 + + * O CO d + K" o 00 CO o 00 CO c] T CI T 1 > 00 O (>« ^ O CI j-^ -^ o i-H CO CO OS lo ■* CO c^ 54 ci -^ +++ 1 + + q + = 1 .*? O re "^ .s -fa to "0 <- Tj< ■>* •* -M O -<1H O O . * Ci 1 -^ 00 t— lO r-< T); q Ol ' ici oi ' o oi + 1 + + + + q CO + a o rH 00 00 *^ i; ■>* uo c: O ■*' -^ o; lO CO CO CI '^ CO 30 o co_ q CO cq ' oi co" co' CO d r-H (>1 r-l + + + + + + a q -q q o CO q O i^ -.* O ci -^^ 05 lO -^ CO !M "* CO d 00 q i>; CO -^ q * oi t~^ d d 00 +++++ + Ci ci + 1 o fa o fa n s & >^ fa o H fa B H < lO q o q CO lO -^ 00 O ■*' CO r-^ O T)5 Oi ■* CO CO iM CO CO CO O v-O >0 O lO o o t- t- (M TJ< tN Ol O o CI c^ - o o CI ci Cl 1 CI ci Cl 1 a V H 2 . > > 00 q 00 q (N -^ -^ i-j t^ t^ ^ -.^ ,-; cc oi CO 1-^ "^ CO i-^ CO -:)-[-. q Tf q q .-; ■ CO r-^ CO d '30 o 1-H d >-< 1-1 -^ O 1 1 1 1 1 1 1 ' 00 CI 1 re 7q o? 00 ■<* q ^ t-^ -^ -^ 'I' Cl CI CI >-l r-l + + + + + + + o id + 01 M O < O H ■pailddB gaiuij oot jfxpajjoa paSpriC sau jad •jojJa JO -jnao Jc J rH * OUR NOTIONS OF NUMBER AND SPACE. 19 w Si 3^1 X) 'M O Tt< O '^ q q * CO O O O CO ^ q (>) q Ti; q 1.0 d q q d 06 — d lO lO 00 o o 00 CO t-^ CO I— 1 ** T— t C5 O 'iH CO 5 ~ • 5 a T-H j'l Q a q 00 1 d) H M ^'l .S «r — dj ^ '^l - - ■>! >a X ■* + -" >^ K 2 «5 *3 a < E-i __l a. -g m iq o bPn s CO o + ta aS O S •« £ a 5 ,^ aa t- q CO q q TJH CO vq Is H lO q lO l>; t-; o O iq o ' CO ci LO ■* t-^ I— d 1 o S p a CO >0 05 OO CD CJ 00 OO lO oi T— ' I-H 1 is C-. O -^ (N 00 t^ GO lO o 1 M + 1 1 1 1 1 »■-£; ^ 2« T^ q t-; q q rH C<) q CO -^ .5 £ ^ •o O O rN t-; iq q O LO "* ■ LO d I-H c^ I-H (?i ^r^ o -^ o "i »-< t-^ (N ^ l-^ (>£( ci Ci O -^ CN 00 oo t~ TJH ■* + 1 1 + 1 1 1 1 1 ')H 0) 4J a ja uo q T-; q 01 •rt; t^ CO q « S? S ?1 t- iq '^1 t-; o o o >-; •«rf lO lO (oi T-i 00 M S _^ !•> O lO 1^ r~^ I-- Ci d to 1-^ 1—1 " o s C. ^ -^ Oi ) -^ q r-; q '* CO GO 1 lo t^ O t^ iq o p "^ T-H ' ci oi d CO •"I" .-! -rJH r^ 1 r-( CO lO i6 ci 1-H Ci OO OO d r^ UO Oi CI ».o CO 00 ■^ Ci O 00 lO i-H — ' ^^- 1 0} a z '3 s S s aj ra a ^ a ^ 0/ fcj 5 tj ~i cs 3 a OJ CD !^ ^ 3^ A s i 5 s g r, ^ cS o CO 000-50 s s £3 ci <» Oi s a ce ci > < P^ ^ l-H o o o -'^ o 3 o O O »^ :C 5', ^ 1- i-i in fn -^ ^ fn fn IJH P^ 1 H Ph P^"< Pt| f^f^ PmP^ •pailddn soim) fK1I jod j -jojja JO IIIM J M EXPEEIMEXT B. WITH pixs SET IX triax(;l?:s axd squares. Apparatus. — Triangles and squares of proper dimen- sions were cut from specially heavy "trunk" cardboard. The pins were thrust through the board at right angles to its surface, care being taken to have their points lie perfectly in the same plane. To the " distance " and " number " categories of Experiment A was now added the category of " figure." The two " figure " categories used in the present experi- ment were those of the triangle and the square. The same " distance " categories were used as before in A; and, as before, these measured the distances between the end pins, or, as it Avould be in this case, measured the outside lines of the triangles and squares. The ''number" categories here used will easily be understood, while referring to the horizontal headings of the tables, if I explain that "HIT" means a triangle with a pin in each corner ; " IV T," a triangle with a pin in each corner and one in the center of the triangle ; u yi i'^'" a, triangle with a pin in each corner and one bisecting each of the three sides ; " VII T," a triangle with six pins arranged as above and still another pin in 22 OUR NOTIONS OF NUMBER AND SPACE. the center. "IV S," ''VS," ''VIIIS," and "IX S " indicate squares similarly arranged to the triangles. Method. — This differed from that of Experiment A only in that the pins were now applied to the subject by some one other than himself, he not being able to handle this apparatus without learning thereby some- what of the size of the board which held the pins. The subject now had three judgments to make for every application. He usually made, and always announced these, in the same order, and as follows, I.e., "distance," "number of pins," "figure." E.rplanation of the Tables. — These tables are much like those of Experiment A, except that a fifth main horizontal division has been added, giving the number of correct judgments as to the figure in which the pins were arranged, calculated from one hundred applications to each person. Also for the better comparison of the result^ from the " triangles " with those from the " squares," a more complicated arrangement of " averages " in the several minor or sub-tables was requisite. This, however, will be clear if, referring to Table 11, I explain that any figures found in the vertical column marked "T." are averages of the foregoing figures, in the same horizontal line, to be found under the four vertical columns marked "Triangles"; and those under "' S.," similarly, are averages for the foregoing figures under " Squares." OUR NOTIONS OF NUMBER AND SPACE. 23 Tables 11, 12, 13, and 14 are " regular "' in method and compare, respectively, with Tables 1, 2, 3, and 4 of Experiment A. Table 15 is "irregular,"' in that the pins were per- mitted to be pressed only three times in succession, as in Table 5. Table 16. Attention concentrated solely on number of points actually felt, as in Table 6. 24 OUR NOTIONS OF NUMBER AND SPACE. o > 00 05 c; CO c; ci ci C5 c; c. CO CO T-H 1-H rH i-H rH 1 1 1 + 1 1 >- O CO C5 o o ^^ Ci Oi Oi o o CO rH rH ci (N -*< (M + (0 10 Fi r' cc c; C' o c: O C5 C' o o c: "^^ CO ^1 + pi h> t^ 00 05 C» Oi 1 1 1 1 K" CO I^ CO 05 Ol cs 05 oj 05 c; CO 07 0? T-H ,-H Ol M 1 1 + 1 1 >; O CL o o o Ci C: c: O C Ci C- J CM T-H + + + "" + r-i H 1 1— 1 CO c; 05 o o Oi o: oi o o T-l rH oi Oi CD CC 01 01 + c T-l rH (M (>i CO 45 rH r-i 5^1 oi CO u a) > < O an Ph CM d !2i saunj 001 jad ^ijoaj -joo paSpnC saun^ -o^ •paSpiiC sat J 'o^ JO jcua JO •;«80 jaj; •SNij >iO Haa MOJ ^ ao SiNHKOanp OUR NOTIONS OF NUMBER AND SPACE. 25 (N p 'TJ O ■* 00 t-^ t-^ ci oi C5 ci oi C5 c; 0* c^ t~ ^1 "-O Ct oc t-^ :d ci c^ 00 c: ~. C-. a c. a CO ^ 1-^ CO ci C; 02 Oi O <35 « i-H rt (M !N ' sami) 001 Jdd X[)33J -joo paSpnt sanii) •o^j; o i-H cc o + " + "l" o o c; o -^ o TTT+T £S + + + O CC o ^ ^ ^ o o + + -F uO p lO « + + + + CO CO ^1 + + + 1— I r-< 3<) l^i CO + •poSpnC aouB^stci JO .lojaa jo •aoxvxsiQ JO sxxanoa.if O r3 be 0} « S ^ > W s •panddB sauni 001 -lad sjuaiu -SpnC ^oajjoo jo 'oji ■aH.iyij JO SXXaKuu.if 26 OUR NOTIONS OF NUMBER AND SPACE. o "ej) S ■pj 4J .a +j P m f^, N 0) •S 1 o n h *i a o o a «5 ^5 ""J Tt* (M 00 O O -^ , lO >0 ?0 l^ CO CO I— i ,— I 5^1 (N CC CC sarai? 001 J8d ^noeJ -.loo pagpnC sauni 'on + + + 02 c0 i-H O CO CO --I i-l +++ + + I 00 O CO (M ++++++ t-H CM (M CO CO + + + •paSpnC suid "OK JO JOjjg JO ■!)U80 jaj •SNijj JO aaaKa^ slo sxKarcoanp OUR NOTIONS OF NUMBER AND SPACE. 27 (N c<; 00 -* o CO t-; o ci i-H od CO CO 00 CO t^ t~ CO t~ Ci t^ t— 'M LO -N -^ CC -* .-H c^i + 1 1 + 1 1 1 + ; CO p 00 p Ci CD CO ci C^ C5 00 Ci Cl Ci C: CD a> !>; CO CO p p ! ■ ' ' + 1 1 + 1 1 p p I- p p p T-H CO Ci d d d CI CI CI CO CO t^ CO t-; (M 00 --H CO --! ^ CO CO t^ CO I-- CJ CD CO ->] t- i-H 10 p 0^ 'N (>i ' rH + 1 1 + 1 1 1 CO CO t^ t:^ t^ t-^ LO t-^ ci d d CO c; CI CI C: c: p CO uo CO t^ CI d CO + 1.8 - .0 + 1— 1 1—1 rH 1—1 CO ^ CO CO Cw t^ I^ CO lr~ CI ci CO UO CO CO + 1 1 M 1 + t^ CO cs CO ci ci 1— 1 i-( r-< CO c; •>! I— 1 00 UO ->! CO ,^ c; CO ,-( r-H ,-1 ,-H CO CO CO CO 0> 01 CO ■* CO .-^ 00 t:~ CO CO t^ Ci CO « + 1 T 1 1 1 Cr5 + Z2- ■ — ■ CO ^ 'CO '30 C: l~ ^ 30 000000 00 06 CO CO t-- CO CO 1^ t- r- i^ t^ i^ Ci CO CO + + 1 + 1 1 + t^ (M rH CO t-C5 CD CO 3D LO I-H ' T-H 1 1 1 7 000 O' t-; 1 CO t-- TJH CO CO r- r-f'* r- CD ^ CI ?N CO I- «o CO I- c: CO --1 + 1 T++ 1 1 01 r^ -rr i-H 'M X r-H ^ 1- t^ CO C5 Ci 1—1 1—1 LO lO CO >-0 ^ CO cooot- CO t~ CO + 1 1 + 1 T + t- r-H -M c: ^ CO CO CO 00 CO t- C-. p CO TtH CO rH p Tt; p "ti ■ CO T-^ ++++ 1 1 LO + ■^ t~ I— ci ci ci 00 I-l 1—1 00 10 lO 1-1 r-! cq c^ CO CO < LO W5 1-H 1-H Ol C^ CO CO ID LO LO LO 1— 1 i-< "M 0 o CO CO + s CO CO + 3 K to E^ r-l O .-H CO CO lO CO d CO ^ O r-; 00 ^ i-H TTT CO 1— I 1 ^ Oi 05 t~ CO t^ -# -^ CD lO ■* i-i CO o o CO T-i ci c^ ,-H cq ■-< --H ■ o4 iM iM (M 1 1 1 1 M T K' >0 r-i 05 CO -r*< O ^^ CO -^ -"i* »o o t^ CO O CO r-l CO (01 '^' d d <6 c^^ n Oi 00 + 4J n 1 1 >; -t* 'M O CO O rH CO i-O -^ CO lO CO 0>] ^ '^ ^ q o CO q oi 00 00 r-^ d i-^ CO (M Ol Ol 1— ' ++++++ Ci CO 01 + 2 P o >o .-H CN 0 lO O i-H CO (M CO ■* lO lO ■* oi ■* (N (01 rN o ■* ■-; rH d CO 00 00 Ol 1-1 ++++++ 00 d + TO H 5 CO -* O O CO (Tl (N CO ■* Ttl Tti O o d c^_ '^_ o^_ ^_ c:^^ o6 lO 00 d d CO lo ■* (oq (M (01 1-< ++++++ CO + 1 f --a lO »C »C T-H .-H (M Sio Haa KllJ sj ^o sxNaKoanp OUR NOTIONS OF NUMBER AND SPACE. 29 O CO O 'M CO 'O t- cr •TO q o o lO t:; r- ■— ' 1^ o ;~ t^ 1- t— t^ c: 1^ CO t~ >1 lO O CO oi o t^ oi CO J— I— t^ o t^ c; o »o> o 1-H i-J 0^ oi CO CO •pajlddtj sauin 001 ■18'J ^1%03J -joo paSpnC sauiij 'Ox; lS> O lO 00 ' + + + + I + CO c; X q r-; i^ OC r-^ rH (>j +++++ 1 CO o4 + b-; q 01 q CO I— _ Oi ■ r-I i-H + + + + 1 1 q + +3.1 —1.8 -2.0 1 o 00 q rH (?^ uo 1-i (>i ' ' CO ++ + + + \ + o o O C --^ t- o C-. ^ O? CO .-1 ++++ 1 1 CO + C CO -* X c; (M Ol +++++ 1 + 3o CO O^ 04 .-1 i-H + 1 1 + 1 1 + ,-; oi X q CO q +++++ 1 oi + 1- o Ci 't :;: oi :^ CO +++++ 1 OI + O O lO ^ ^ ^ (?i CO 00 -i; •paSpwC saotre^sjci jo joajg; jo ■%a9o jaj: •aoxvxsi(i JO sx.N;aKoa;ip O tN O Oi (N O o CO CO q 0^1 o CO :o lo 00 ci 00 O O t~ t-; 01 lO 00 CO CI Ci c: O O L.O o 1 1— H o^ oi CO CO •paiiddu S9uir} 001 -lad jS(103J -JOO pa3pnC sauiii -o*! •aH.ioij ao sx^aKOU-ix" 30 OUR NOTIONS OF NUMBER AND SPACE. 1 o Q m < "A O i < CO ci + 41 <5 © 5i CO CO" + 2 ni 3 C5 + '^ g^^^^g CO CO t- 00 CD in •^ lO t-^ 00 Ci 00 TT 1 1 1 1 © 1 •n a 1^ coo."]i-icoTtoco © CO O IT] © t-; C^l C^ C^l ocoo6i-^si'*»io©© TTT 1 1 1 1 © + ^ CO (N tC C-] 01 © © © 00 t-; r-i rH r-I LO -t C-i 't -* oi ?] i-H rH C-1 ml + 1 + + + + + + 1 n ■4-) a 0) !"• i """ic in »q © © lO © lO iq Oi'l^in-OCO-Tt^OOCOrji r-(T-li-ICO'** CO t); CO in in i-^ © ^ ci Cl O-I i-H C'J CI C-l 1 1 1 1 1 1 in ci 1 r' ^ CI •>* CO c~j iM w e-1 00 C0t-;COt-;CO©i-;TH© ci -^ ci rH co" "* ci co' <* TTTTT i^TTT 00 1 (MiHOtOCOOOrtiT+lcc © ©inioin©in©in© CO c-i CO ci ci -o r-i i-J in T-H cq rH rH 1 + 1 + + 4- + + + CO + H H S CO >* © » •^_ X © CO © CO CO oi 'I' -)< in -^ i-H c5 in co' CO CI 01 -t< in ;o CI oi i-H +++++++++ CO CO + Ed >o lo ira »o iH rH C-1 CJ CO ^0 •* Tli lO a) in in in in rH r-! C^ Ci CO co' ■* •<* in < o 5r, c c sauii; 001 jad Xnoaj -.loo pgSpnf sauu; 'Ofj •paSpnC siud 'oa JO jo.ua JO •?u8a aaj •SMij io aaai\ .a^ ; do sxKaKOuiif OUR NOTIONS OF NUilBER AND SPACE. 31 1- L- X x k- — ri o m z: X -^ r! d d '^ ^ •* <£ H ■■c -.c --^ -^z »-•; lO'-c ~ X --c ~ X X -^ t~ -f Lt ~; — ; + 1 1 + 1 + 1+1 1 X q L-; X i"^ X ■#. ?: 11 S 11 "1 o q t-; L-; L-; ?i c^] iq ti — < •.£ «* ci IS c; id •^ »r; r-. r. :- -r q -^ '^. •* >>: t-^ -«• ' C-i IT i-i r-1 ^ + 1 1 + 1 + + + 1 C-l + yt£sg|g8g X r^_ q c c rt ir; c o t-; d t-^ id 1— t-^ -r t-^ cc' id t- 13 -^ -.c L"; I--; o :C t- ^ ri ct-;r:xqaqq'*q c-i tt rd ■ c4 Ti' r-i ' e-i + T 1 + 1 + 1 + 1 q 1 •I ^. "^ ^. L* «3KEHg g rf CC X CI .-; c^ c-i rt ?i c-i cd c^ + 1 + + + 1 + gggggg g £5SS":h:;SS:3 X x + 17.0 -10.0 + 2.0 + 7.4 + • 7 - 2.2 + gg?:SS8||| 3 ^5ss?2§-^g^s X Or5iC?OCO-*(MT-IC<5 S-+ 1 1 + 1 1 1 + 1 + gtrSggggSg q id i2S^^SS5E=? - ^^q ^: o q q •* o •* q -* .-; ?i tt rd 1-4 ■ ■ r-! + 1 1 + 1 + + + 1 + g?.£S|||S| i SSSis^S X CI ?^ t- X O C*l 7-; 1-; r-j ■ c-i cd 1 1 1 1 + 1 q T x s 5 c c s t-; SgfcggS^sS S3 q cc q T); q X c^ c? q •* id j4 c5 •* e^ ' " c^ + T 1 + 1 + + + 1 1 --■^•lllils i i!Sd3;fS^l5I=t3 X 4 q -^ o q q o t-; < lO O lO o T-l .-; -M f i rO cd "* Tj^ L- u V > < 1- o o 1.-5 1- r^ M ~i r: ^: ■* •* 13 m 1 u > < sauiii 001 Jad ^naaJ -joo pa3pnC satuii -ox ■pagpnC OOUB^SjO JO JOJJ^ JO •p3![ddB -juo paSpnC saiuii -ojj •aoKvxsi a JO sxNaKoaaf io sxiiSKoaap 32 OUR NOTIONS OF NUMBER AND SPACE. K <1 H K O Em >> •g 3 » o S 0) +-> s > a CR (U IH Oh 1^' W 3 a 11 I* <1 o5 OJ T a; CB u to > < o S T 01 01 + 02 y. 05 r-i q 00 ooo CO t-; 1 1 1 !0 !> - 1- CO c: t- o ■-1 'N Ol .-^ oo o (>1 q q q o oo ci c c; t~ lo c. fM , — . 1— 1 (M ■<*( c; o o 0-1 O (N Tj< 00 1— 1 I-^ CO O -*■ CO' 2 T 1 1 M q 1 (U If] ai > 5i t^ >0 t^ 00 o ^ O-J CO -* Tfl 'CO ^ q O. o o o c c; >d c; o + + + + + q + 1. 1 H; CO o q 00 q -l T-H 00 O Ci -* •* S cr o r~ 00 o OO CO 1— 1 oo -* X 00 O O 00 0-1 t^ r^ -^ CO" oo' 00 ■*! Tti O C^I + + + + + d + 3 T-i --; T-< i-H 0>1 1 r-H o o o o o GO >-o ++TTT + O t^ O ffl ^ cc 00 !M — 1 r-l O CO .-H r-l » CO + 1 1 1 1 + oco o -* --1 -* ■* lO O -^ GO Ol ■* i-H rH ++ 1 1 1 + o o o CO o o Ol X "w -M CO f—t CO 1—* T— < + + 1 M + 1 1-i oi oi ■ ■pa2pnC 8DUB}S!a JO Jojjg; jo •aaxvxsiQ jo siKHivoa.if t^ O O 01 Ci x> lO iQ lO lO lO o CO 01 01 0) -o J-. o ^ cr. CO CO "^ "^ "^ o -*l CO CO CO 01 o ^ --0 r-l O C2 Ol iC UO <» 03 s T-H 1-H Ol Ol CO •pailddB saiun (X)l Jad Xnoai -Joo pd3[nif soiui; x 34 OUR NOTIONS OF NU]MBEE AND SPACE. s <; 0) q + a; s o id + id S CO 00 1 a" 1 ^ oo o q r- + 1 + ' ® s B > o o o o o o q q q Gc o ri CO :n ' i-h' t— •N i-H CI + + + 1 1 o 00 + > _ ooooS q q o o 00 or c oo ^ o o o d CN CC 1 1 o 00 1 1 > o o o o o o q q q rN q -^ o' ao' >o .-! ■^ CO i-i cq + + 1 1 t d + J >> O O O 00 -M q (Z>^ <^_ ; q 00 q d d -^ 00 oi lO ^ i-i + + 1 1 1 CD + O OS H a O Q W i-H i-i oq (>i Of 0^ > --1 i-i tM c o •S «r H o ■4-> H H of 3 +j ta a o W) (D o a is (« ■u § XI 01 (U 00 nS o \ to in o H ?; . oo gQ i^ ^ 0^ '^-' ^J <^ • ^6 H tc oi (» & 2« W « W (> C O !- t-; CO p 'O iq p CO O ->! r-I O Oi CI O TjH -* fN I- >0 CO o CO C; -*l tN CN i-H ■ uo CO ■ d t-^ 1 +++ 1 + + CO o o p ic p CO d o o o T-H C-. 1- -* -H iM --• d C^ p CO CO CI p ' d r-i CO d >.-f 1 +++T + + ai t ^ ^5 Ol O lO p 00 --! i-^ CO CD ao Ci lO CO CO (>» CO >0 p p CO CO CO lO lO 1-5 d 1 + + + + 1 1-; CO + H K -< c y, O O CO 1-; p CO .-I ci d CO o —. l^ CO "* (M CO CO S-; p cq (>j ' d 00 d i>^ <>i ^++++ 1 + 1-^ + r- h^ ^ lO (M I— 1 -"Jl p d oi d d d d O CO -* lO CO CO T-H O] p P P O " d CO d (>i d Cl CM 0-1 --I i-H ++++++ P I>5 + 1 K ►J w !> o r~; CO p p r-^ d lO d CO o Ci ^ CN CN CO CO CO CO CO lO -^ o ■ d d ci d 00 O) ^ 1 1 1 1 1 1 T r; C<) t^ CO CN GO CO d CO 1-H d o c: ^ -^ -^ oo CO 01 01 -O CO Ci 1-1 ,-1 1-1 (M 1 + 1 i k + 05 1 a r- -* t^ lO p p p d d o^i CO lO c-i C~w lO TjH -* 00 i-i c T-H ^ CO t-; p p ■ rH d CO I* C4 1— 1 r-l i-< + + + + + + + H s CN CO p -^ 0O_ p d d d CO T-i d O vC ^ -^ OO 0^ CO CI p t-; p -* p ' t-^ 1-H CO d lo 5 1 o o o -^ o o H flH plH -< PH P4 0) to R,^ rt o "^ «* q ^^ !h ^ !h ^ o o o -^ o o Eh Ph Ph "s Ph Ph 0) > < •pailildv! S31UH 001 Jad •paSpnC aDuujsiQ }o aojag JO •juda aoj •SNij ^o aaa I\[Q sj JO sxnaiMoanf OUR NOTIONS OF NUMBER AND SPACE. 37 lO t~-M C-. CO 5^ c; t- t^ OJ CO l^ C: O o -# o CO 00 c: 1^ O O c; r~ t^ « ■* .-H OO (Tl Tt< T-H (M aa t~ iS C t— t:- :s CO o C t~- O -M O oc -M t- rr X i^] CC I-; CC C; CO X c; x' r: "' Ci X t— w o c: r^ cr r-; o t-^ :f ri x' ^ X ri r-; O cc t-^ i~^ t-^ '>! c; C; t:~ l^ O iM ^ -M ut; X O O p cc « O CO ^ O -1" 00 C. Xj t^ '^ ^r + + + I + CO lO O CI o * r-5 CN 1-i i-H + + + I + C .-H -M r-H CO 1 1 ++ 1 1 ■* 30 lyj t-; +++ 1 + .-H :c --r> o o ■ r-H >-; to 1 1 + 1 1 7 cc -M C^ IM O 1 + 1 +T o CO 1 .-; c^ Ofl p CO T-H ' cc I-< t-^ 1 ++++ + ^.^ o cc c: cq cc cc ++++ 1 + O -* Ct p ^ r-; 1-! ' 1-^ X 1 1 + 1 T 1 p r-; lO lO ■^_ 1 1 + 1 + 1 CO CO !M_ X O t~^ ■* O" '^' + ■^ P '-^ ■"! O t-^ -^ O (N o ci ci c: n o o O O C: O O o o c; o o cc (N cc X cc c o o X cc uc ^ cq O '■ CO O tH CO r~ cc t— O O c: CO c: X IM CO :0 O O i_C w ■M X o i~ -^ 3b' ^ a u o o rt o =5 5 = ^ o c ^ < t, &L< i t< t-l c o o ■^ o o E-i fs, flH ■< PH pIH 'paSpnt a3a6)si0 jo jcujj jo •JUSDJ.IJ -p3I|ddB 33U11) 00[ J9d Xi)3^03 paSpiiC saoii) -ojj •aoKVxsiQ JO sxxajvoajp JO sxNaKoaap 38 OUR NOTIONS OF NUMBER AND SPACE. o OS ^ a CO ^ o » K H 1^ e H < o ;?; r« m lO <1 --1 J -:! K ?; ,_t M a ^ OQ pj H g !*; c c o o -^ O ■* lC O (M fN o r- '>q CO o o ■* ■^ <© ^ ,_ ,—1 ^H o o S S S G S §% 5 s 5 B o o o -5 o o H 6m tM •< Pm Pm •pailddB sauH} full J*J CO » CO ' T-I T-; d t +++T 0 ■ -< r-^ d + I + I ! Ol C". Xi X ^ ' d c-j I + I + + + + + + -o CO CO Oi CJ p CO CN O oi r- oi -^ O Oi Ci Ci o CO 8 CO >-•:> o o O CO >0 CO ^ O Ci Ci O Li Ol CO r~ iM p T^ O CO ^ t^.-H O C5 O CO lO CO (M O CO t^ o ci ri -^ l-^ O CO CO t^ o o CO Tongue Forehead Forearm Abdomen Forearm (a) Forearm (b) 0) ci u IP ■paijddo 9.1IUII oot Jnd .CHO.UJOD paSpnC sauiij ofj ao sxx3iv:>a.i [* - EXPERIMENT C. WITH LINEAL FIGUKES. Apparatus. — The lineal figures were made from cardboard of medium thickness, but very liard and strong. This material was chosen to avoid temperature complications. Great care Avas taken that the lines, and particularly the corners of the figures, should be perfectly even, sharp, and accurate throughout. The importance of this cannot be fully appreciated unless one has acted as subject for a long period. It is little less than marvelous how slight a cue will be noted by which to remember a particular piece of apparatus as " tliat same old one," and so the judgments become based upon a fund of past experiences and imaginations, rather than upon a new and present impression, as is absolutely necessary for the work here in hand. In our work, if any piece became thus "■ individualized," it was at once discarded. The figures were made like deep pasteboard boxes, with one end (that to be pressed on the skin) left open, as when the lid of the box is off. The larger pieces were braced, as it were, with false bottoms, one or more, as needed to make them firm. OUK NOTIONS OF NUjSIBER AND SPACE. 41 The figures used -were triangles, squares, and circles. The categories of " distance " remained the same as in the previous experiments. There were no longer, of course, any ''number" categories to be observed. Method. — This was precisely the same as in Experi- ment B, but a new care was required in applying the apparatus to the subject. The pieces being made hollow, like a drum, they would, upon the least slipping of the fingers over their surface while handling them, give out a sound Avith the spontaneity of a resonance box or a tambourine. This sound would become individual- ized by the subject for eacli particular piece, the same as a bent corner or an imperfect line, and, in a way making the judgments worthless if such sounds were permitted. The utmost care, therefore, was used throughout, in handling the pieces, to avoid every particle of slipping or rubbing of the box, either upon the subject's skin, or upon the fingers of the operator. The tongue was no longer investigated, as the appa- ratus now was too large to work with comfortably in the mouth. The Tables 18a to 22a, inclusive, for Experiment C, and 18b to 22b for Experiment D, will be understood, after examination of the similar ones for Experiments A and B, without further explanation. These above- numbered tables of Experiments C and D correspond, respectively, to Tables 2, 3, 4, 5, and 10 of Experiment 42 OUR NOTIONS OF NUMBER AND SPACE. A, and to Tables 12, 13, 14, 15, and 17 of Experi- ment B. Table 23. This table shows the distribution of the whole number of figure-judgments. They are calcu- lated, in per cent., from one hundred applications of each piece of apparatus to each of the several regions of skin worked upon. For example : the three numbers, 65.0, 21.0, 14.0, arranged vertically in the upper left-hand corner of the table, mean that of the total number of times that the "1 centimeter " triangle was applied to the various regions of the body, in 65 per cent, of those times, this triangle was judged correctly to be a triangle ; in 21 per cent, it was misjudged to be a square ; and in 14 per cent, a circle. The purpose of this table (to be discussed in our general study) is to aid in comprehending the errors made in judging the figures. , The heary figures in this table show the correct judgments ; the other figures show the false judgments. EXPERIMENT D. WITH SOLID FIGURES. Apparatus. — Like that for Experiment A, except that the pieces were made of cork, and solid throughout. Method. — Precisely that of Experiment C. Tables. — See ''The Tables'^ (page 41) under Experi- ment C. 44 OUR NOTIONS OF NTJMBER AND SPACE. Table 18a. Hxperiment C. — With lineal figures. FOREHEAD. Person Distance (Centi- meters) Averages of N. and P. Figure Triangles Squares Circles Averages of T.,S.andC. O < H m No. times judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 86 52 56 50 42 48 (226) 32 50 32 30 44 88 72 54 64 38 48 62 63.3 52.0 50.7 39.3 44.7 66.0 En O H Averages 55.7 46.0 56.3 52.7 Per cent. of Error of Distance judged. 1 1.5 2 2.5 3 3.5 + 7.0 -4.0 + .5 - 1.6 -1.0 -8.2 (229) + 42.0 + 20.0 + 28.5 + 17.6 + 15.3 — 1.7 + 16.0 + 12.6 + 5.0 + 4.8 - .6 - 6.7 + 21.7 + 9.5 + ,11.3 + 6.9 + 4.6 - 5.2 Averages - 1.3 + 20.3 + 5.2 + 8.1 Cm O Q •-5 No. of correct judg- ments per 100 times applied. 1 1.5 2 2.5 3 3.5 74 70 68 60 72 54 (232) 32 46 64 72 78 82 54 38 54 86 80 84 Averages 66.3 62.3 66.0 64.87 OUR NOTIONS OF NUMBER AND SPACE. 45 Table 18b. Experiment D. — "With solid figures. FOREHEAD. Person Distance (Centi- meters) Averages of N. axd P. FiGCRE Triangles Squares Circles Averages of T.,S.andC. o H No. times judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 52 52 22 22 34 84 (235) 34 30 26 28 28 80 56 44 40 14 42 78 47.3 42.0 29.3 21.3 34.7 80.7 O H o a Averages 44.3 37.7 45.7 42.6 Per cent. of Error of Distance judged. 1 1.5 2 2.5 3 3.5 + 32.0 + 2.0 + 22.0 + 11.6 + 6.3 - 2.8 (238) + 53.0 +38.5 + 40.0 + 21.2 + 8.3 — 3.4 + 29.0 + 25.3 -17.0 + 12.8 + 3.0 - 3.1 + 38.0 + 21.9 + 26.3 + 15.2 + 5.9 - 3.1 Averages + 11.8 + 26.3 + 14.0 + 17.4 >£4 o No. of correct judg- ments per 100 times applied. 1 1.5 2 2.5 3 3.5 74 68 62 76 82 88 (241) 40 58 68 82 92 94 54 6(5 84 82 86 94 Averages 75.0 72.3 77.7 75.0 46 OUE NOTIONS OF NUMBER AND SPACE. Table 19a. Experiment C. — With lineal figures. FOREARM. Person Distance (Centi- meters) AVERAGE.S OF N. AND P. FiGUKE Triangles Squares Circles Averages of T.,S. andC. H No. times judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 44 36 40 38 20 12 (244) 32 28 38 26 38 76 30 30 42 32 48 28 35.3 31.3 40.0 32.0 35.3 38.7 \n Averages 31.6 39.6 35.0 35.4 H a Per cent. of Error of Distance judged. 1 1.5 2 2.5 3 3.5 + 40.0 + 24.6 + 11.0 - 1.6 -18.7 - 23.4 (247) + 54.0 + 34.0 + 22.5 + 9.2 + 3.7 - 4.0 + 71.0 + 12.7 + 10.0 - 5.2 - 7.3 - 18.0 + 55.0 + 23.8 +»14.5 + 1.5 - 7.4 - 15.4 Averages + 5.3 + 10.7 1 + 10.9 + 12.0 o No. of correct judg- ments per 100 times applied. 1 1.5 2 2.5 3 3.5 62 58 68 44 38 26 (250) 56 48 48 54 56 76 40 44 58 48 60 82 Averages 49.3 56.3 55.3 53.6 OUR NOTIONS OF NUMBER AND SPACE. 47 Table 19b. Experiment D. — With solid figures. FOREARM. Person Distances (Centi- meters) Averages of N. and P. Figure Triangles Squares Circles Averages of T.,S.andC. w < No. times judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 38 32 38 14 58 38 (253) 30 30 32 24 50 80 40 26 30 32 54 80 36.0 31.3 33.3 23.3 54.0 66.0 U* Averages 36.3 42.0 43.7 40.7 H d Q Per cent. of Krror of Distance judged. 1 1.5 2 2.5 3.5 + 53.0 + 9.3 + 14.0 - 9.2 - 6.0 - 13.4 (256) + 65.0 + 36.7 + 22.5 + 26.8 + 6.3 - 4.0 + 52.0 + 26.6 + 4.0 4- 8.8 - 2.0 - 3.4 + 56.7 + 24.2 + 13.5 + 8.8 — .6 — 6.9 Averages + 7.9 -f-25.5 + 14.3 + 15.9 o a 2 O Cm No. of correct judg- ments per 100 times applied. 1 1.5 2 2.5 3.5 52 56 44 36 (i4 54 (259) 36 44 38 38 68 68 42 56 (i6 62 80 88 Averages 51.0 48.7 65.7 55.1 48 OUR NOTIONS OF NUMBER AND SPACE. Table 20a. Experiment C. — "With lineal figures. ABDOMEN. Person Distances (Centi- meters) Averages of N. and P. Figure Triangles Squares Circles Averages of T.,S.andC. < H No. times judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 50 36 36 30 30 36 (262) 48 32 26 40 40 76 42 46 32 42 40 34 46.7 38.0 31.3 37.3 36.7 48.7 . h Averages 36.3 43.7 39.3 39.8 a o Per cent. of Error of Distance judged. 1 1.5 2 2.5 3 3.5 + 43.0 + 19.3 - 1.5 - 3.6 - 10.0 - 17.7 (265) + 44.0 + 17.3 + 28.5 + 13.2 + 6.6 - 5.4 + 52.0 + 18.6 + 1.5 + .8 - 3.0 - 13.1 + 46.3 + 18.4 + ' 9.5 + 3.5 - 2.1 - 12.1 Averages + 4.9 + 17.4 + 9.4 + 10.6 O 1-5 No. times judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 52 72 48 74 56 46 (268) 38 52 46 60 72 78 36 64 86 82 92 80 Averages 58.0 57.7 73.3 63.0 OIJR NOTIONS OF NUMBER AND SPACE. 49 Table 20b. Experiment D. — With solid figures. ABDOMEN. Peksox Distance (Centi- meters) Averages of N. a:sd P. Figure Triangles Squares Circles Averages of T.,S.andC. o 15 ■< H to No. times judged cor- rectly per 100 times applied. 1 1.5 '> 2.5 3 3.5 34 24 54 26 32 48 (271) 44 28 30 32 38 74 46 32 52 34 48 68 41.3 28.0 45.3 30.7 39.3 63.3 O o s Averages 36.3 41.0 46.7 41.3 Per cent. of Error of Distance judged. 1 1.5 2 2.5 3 3.5 + 55.0 + .6 + 2.5 -12.0 - 5.3 -15.7 (274) + 48.0 + 26.0 + 27.0 + 11.2 + 6.3 - 4.9 + 36.0 + 26.6 - 1.5 - 2.0 -10.0 - 6.2 +46.3 + 17.7 + 9.3 - .9 - 3.0 - 8.9 Averages ■+ 4.2 + 19.0 + 7.1 + 10.1 64 o s 2 3 1-5 No. tinies judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 58 62 50 68 66 52 (277) 34 42 46 62 m 72 46 42 62 70 72 90 Averages 59.3 53.7 63.7 58.9 50 OUR NOTIONS OF NUMBER AND SPACE. Table 21a. Experiment C. — "With lineal figures. (a) FOREARM. Pressing evenly three times only. Person Distance (Centi- meters) Averages of N. axd P. Figure Triangles Squares Circles Averages of T.,S.andC. < p No. times judged cor- rectly per 100 times applied. 1 1.5 2 2.5 3 3.5 54 30 48 24 24 22 (280) 30 26 36 14 52 54 30 26 50 16 44 22 38. 27.3 44.7 18.0 40. 32.7 p^ Averages 33.7 35.3 31.3 33.4 03 H o p 1-3 CS jj. to :: O.I' o 1 1.5 2 2.5 3 3.5 + 41.0 + 12.7 + 3.5 - 7.0 - 13.6 -21.4 (283) + 62.0 + 27.3 + 21.0 + 10.8 - 2.0 - 3.7 + 68.0 + 22.7 + 7.5 + 4.8 — 0.6 - 18.0 + 57.0 + *0.9 + 10.7 + 2.7 - 7.3 - 14.4 Averages + 2.4 + 19.2 + 13.1 + 11.6 (4 O T •r a. 'S 2 -d o g 1. c -S . s o S 'A 1 1.5 2 2.5 3.5 72 00 54 52 50 39 (286) 36 38 96 50 60 60 48 50 52 56 52 66 Averages 53.0 50.0 54.0 52.3 V ■ - - ' OUll NOTIONS Or^ NUMBER Aim. SPACE. 51 -■ -if: ■ >' Table 21b. Experiment D. — With solid figures. (a) FOREARM. Pressing evenly three times only. Pkrsox Averages OF N. AXD P. Distances (Centi- Averages Figure meters) Triangles Squares Circles of T.,S.andC. H (289) ^5 1 54 44 22 40.0 ^S^' 1.5 23 22 24 23.0 H O 2 30 18 32 26.7 Z ," 3 32 52 50 44.7 P o « 3.5 20 80 46 48.7 b Averages 34.5 40.3 34.0 36.3 (292) 1^ 1 + 4G.0 + 49.0 + 64.0 + 53.0 S ^ -3 1.5 + 9.3 + 40.6 + 31.3 + 27.1 O O M 2 + 8.0 + 33.0 + 21.5 + 20.8 >-s 3 2.5 - 1.0 + 14.4 + 7.2 + 6.7 3 -11.0 + 8.3 - 5.3 - 2.7 o 3.5 -1G.3 - 4.0 - 3.4 - 7.9 Averages + 5.7 + 23.5 + 19.2 + 16.1 Eiu "2 2 (295) O "5 1 46 44 40 « 2 -s' 1.5 42 36 32 « r; t- u -; 2 40 28 44 s 2 b^S 2.5 40 30 52 ® a o 48 50 36 >-i 3.5 24 GG 60 Averages 41.0 44.3 44.0 43.1 52 OUR NOTIONS OF NUMBER AND SPACE. Table 22a. Experiments C and D. — Witli lineal and solid figures. SUMMARY. — Averages brought forward from Tables 18a to 21b. Person. Average of N. and P. Tri- angles Averages Figure. Squares Circles of T. S. and C. >% (298) 1 -d Forehea.. j ^^f > 55.7 44.3 46.0 37.7 56.3 45.7 52.7 42.6 S a T^ I Lineal 31.0 39.6 35.0 35.4 : ^ Forearm j g^^.^^ 36.3 42.0 43.7 40.7 CD CC T^ , s ( Lineal 33.7 35.3 31.3 33.4 3 2 Forearm («) { g^^j^^ 34.5 40.3 34.0 36.3 m All ( Lineal Abdomen j g^,j^ 36.3 43.7 39.3 39.8 % ^ 2 36.3 41.0 46.7 41.3 H . ( Lineal Averages j g^jj^ 39.325 41.15 40.47 40.30 fi o "A 37.850 40.25 42.52 40.26 U o Average of Lineal and Solid 38.590 40.70 41.50 40.28 (301) Z m Forehead i^^f - 1.3 + 20.3 + 5.2 + 8.1 S S 2 + 11.8 + 26.3 + 14.0 + 17.4 t 2 s ,. ( Lineal iorearm j g^jj^ + 5.3 + 19.7 + 10.9 + 12.0 w !• + 7.9 + 25.5 + 14.3 + 15.9 ■s^ ,, / > I Lineal iorearm(a)Jg_^jj^^ + 2.4 + 19.2 + 13.1 + 11.6 ^ « + 5.7 + 23.5 + 19.2 + 16.1 ^1 Abdomen \^^^-' + 4.9 + 4.2 + 17.4 + 19.0 + 9.4 + 7.1 >+10.6 + 10.1 Averages j ^^T^ + 2.8 + 19.1 + 9.6 + 10.5 o + 7.4 + 23.6 + 13.6 + 14.9 Average of Lineal and Solid + 5.1 + 21.3 + 11.6 + 12.7 to (304) a . a .s -r^ , 1 ( Lineal Forehead -^ „ 1 • 1 ( Solid ()6.3 75.0 62.3 72.3 66.0 77.7 64.9 75.0 rg' a F„,.ea™, \^^ 49.3 56.3 55.3 53.6 )^ .ai* 51.0 48.7 65.7 53.1 o ., CO ^ , . ( Lineal 53.0 50.0 54.0 52.3 to ll Forearm («) j g^j.^^ 41.0 44.3 44.0 43.1 [2J 9. ^ . 1 , ( Lineal 58.0 57.7 73.3 63.0 1 o u " Abdomen } g^j.^^ 59.3 53.7 63.7 58.9 t-5 Averages { ^^^^^ 56.65 56.57 56.57 54.75 62.15 62.77 58.45 58.025 Average of Lineal and Solid 56.61 55.66 62.46 58.20 OUR NOTIONS OF NUMBER AND SPACE. 53 (C ^ X 3c s (>i cri o o oi «D ci 1-; X ad t-^ (m' o r-i to o c" + + ci + H CM 2 ei CO H 2 CO IN O O 00 -O -N ac 3D r: o X rr o CO ^ T}i o to •* bS 0<) i-H -^ p ■^_ p r-H p lO CC O O -^ t-^ "I^ CO 1 1 T 1 T 1 T 1 X t- ,-; to 1 1 OS 1 t-; t-; CO p O t-; t-; or •<1" -* ut" ^ o '^ l5 c: ■<*i cc OT u" -* •<* :c 'T 5: ^ pc:^wcct— r-c; ■* o t~^ ' t-^ '>5 ?ci CC + + 1 1 1 M 1 T 1 p T 3 B o ^ CT ?? O 0? p t-; or r- o c: r4 •m' n X. Tf I—' o ^ ec ?q r: ?q .-( ct ct cc J5 ^C:(MOCOt^I:~i-OC5 2 O id .-<■ CO CN O CO p -^ CC t-^ + + o id + t— ^T O "* t^ 1^ CC tC o ci o rr -*' ■-£ r^' L.-" O •M TP ct -* I-J cc ■* ■<»' ft CO CC L.C LC t-; CO O CC + 1 J3 O p CO C'J It p p p r>i •>! ^ ,-; t^ Tt' 30 CC O ■* CC Ot 0-J 3^> Ct !M l~-^ T-H uC p X_ -M p .-H ■* t- C: --I CC •^' O t-^ X i>^ c^ CI CI c^ cq .-^ T-i ++++++++ X ji + + + a OJ to (C )H 0) > < cocoorooot^co c? t-^ o :2 CO o o ^ CO iM r-OOt~OOCCCC t-J X o :c t-^ CC o to' CI CC LC LC uC lC -* Tf ++++++++ p o o x + + + "5 — "? — "^^ "?-- 1-3 X .^ X '-2 X f2 X -^ ^ "^ c -. Jh Sh - c o o -^ fc^ £=^ t. <1 a u 0) > < hi > i3x 2 -3 3 1 -< K < 'i 3 3 1 1* *-! '-I 3 d c3 rt 5 •siuaniSpnp aotreisia Jo JOJJa JO -^UaD J3d^ HJKV xbi(j; j< > sixaKyuax' 54 OUR NOTIONS OF NUMBER AND SPACE. Table 23. Experiments C and D — (For explanation of this DIST.US'CES. 1 1.5 ■2 Figures. T. S. c. T. S. C. T. s. c. (307) (308) (309) Exp. C. Triangle 65.0 41.0 24.5 65.0 36.0 2G.0 59.5 .32.0 13.5 Square 21.0 40.5 31.0 28.0 46.0 25.0 32.5 53.5 24.0 Circle 14.0 18.5 44.5 7.0 18.0 49.0 8.0 14.5 62.5 (014) (315) (316) Exp. D. Triangle 57.5 45.5 24.5 57.0 35.0 25.5 50.5 ,33.5 20.0 Square 27.0 38.5 30.0 27.5 45.0 25.5 32.5 45.0 16.0 Circle 15.5 10.0 45.5 15.5 20.0 49.0 17.0 21.5'' 64.0 \ OUR NOTIONS OF NUJklBEE AND SPACE. 55 With lineal and solid figures. table, see page 42.) 2.5 3 3.5 Averages of all Distances. Total Aver- T. s. C. T. S. c. T. S. C. T. S. c. ages. (310) (311) (312) (313) 57.5 32.5 10.0 21.5 59.0 19.5 4.0 28.0 68.0 54.o' 16.0 38.0 66.5 8.0 17.5 11.5 17.5 71.0 39.012.5' 5.0 44.0 74.0 17.0 17.0 13.5 68.0 56.6 32.6 10.6 26.5 56.6 16.9 14.1 23.7 62.1 L58.5 (317) (318) (319) (.320) 55.0 28.5 16.5 30.5 45.0 24.5 14.5 19.0 66.5 65.0 31.0 4.0 13.0 70.5 16.5 10.5 21.0 68.5 54.5 38.0 7.5 19.0 75.0 6.0 5.0 12.0 83.0 56.6 .30.7 12.6 29.4 53.1 17.4 16.6 20.6 62.7 1 V ^ EXPERIMENT E. WITH A MOVIXG PENCIL. At a certain stage in our studies, we shall have to inquire wliat part each of several elements, which we know enter into the formation of every judgment, individually plays. Important among these are "mass,"' both of stimulations and of feelings ; ''intensity," both peripheral and central ; the '' time rate " of stimulation, and of mental response. Particularly we shall wish to know the separate influence of each of these factors, in order to comprehend their united action in producing a judgment as a whole. Experiment E, still pursuing the comparative method of investigation, has this exigency in view. It differs from all the foregoing experiments by introducing motion over the skin. The apparatus and the method were of the simplest kinds. The pencil was of ivory, 5 millimeters in diam- eter, and rounded hemispherically at the end. It was always kept at the skin temperature, was held vertical, and applied by an assistant. The region to be worked upon was laid out in squares by dots of ink one centi- meter apart. The pencil was always drawn in the same direction, i.e., horizontally on the forehead, and down 58 Oim NOTIONS OF NUMBER AND SPACE. on the forearm and the abdomen. Four categories of motion were investigated : Quick and Heavy ; Quick and Light ; Slow and Heavy ; Slow and Light. The "quick" and " slow " movements were timed by a metronome, until we had, by continued habit, well acquired the beat. ''Quick" was at the rate of about 20 centimeters per second ; and slow about 2 centimeters per second. No attempt was made to gauge the pressure exactly. " Heavy " was as liard as could be borne for a length of time without pain. "Light" was as light as could be distinctly and evenly felt. The tables are so like the other tables that little further explanation will be needed. As only two sub- jects Avere available at the time the experiment was performed, a double number of applications was made, and each person and their results kept separate, as is shown in the tables. OUR NOTIONS OF Nr:MBER AND SPACE. 50 Table 24. Experiment E. — "With a moving pencil. FOREHEAD. Peksox. Distance (Centi- meters) AVEKAGE OF N. AJJD P. Mode of Motion. Quick and Light Quick and Heavy Slow and Light Slow and Heavj' Averages O g (323) •d ~ 1 96 96 94 92 94.5 udge loot lied. 2 82 86 80 80 82.0 H 3 74 82 80 76 78.0 is s s.| 4 56 64 72 62 63.5 l>: 5 68 78 76 56 69.5 1^ 6 90 94 90, 94 92.5 Averages 77.67 83.33 82.0 76.67 80.0 rr. (326) 'A 3! 1 -1- 4.0 + 4.0 + 6.0 + 8.0 + 5.5 ■e 'S -• 2 -1- 7.0 + 1.0 + 4.0 + 8.0 + 1.5 5 <" iC 3 -4.6 + 1.3 + 7.3 + 1.0 •-5 4 - 6.5 + 6.0 -1.0 + 6.0 + 1.1 t-^ C 5 - 1.6 + 1.2 + 2.4 + 0.4 + 2.1 c 6 -2.0 -1.0 -1.6 -1.0 - 1.4 Averages -2.9 + 1.9 + 1.8 + 5.8 + 1.6 60 OUE NOTIONS OF NUMBEK, AND SPACE. Table 25. Experiment E. — "With a moving pencil. FOREARM. Pebson. Distance (Centi- meters) Average of N. AND P. MODK OF Motion. Quick and Light Quick and Heavy Slow and Light Slow and Heavy Averages 8 I (331) •^■9 1 91 88 74 61 78.5 ■^s-g 2 55 77 53 52 59.25 H 3 60 56 48 47 52.75 S5 gl| 4 38 46 39 35 39.5 H g.^ 5 50 46 42 43 45.25 P O H 1^ 6 . 42 80 29 63 55.0 Averages 56.0 66.5 47.5 50.2 55.0 (336) 1 + 9.0 + 13.0 + 27.0 + 47.0 + 24.0 a 5 ■«■ 2 - 8.5 + 7.5 + 7.0 + 21.5 + 6.9 K> " o .5? 3 -10.0 + 11.0 + 2.6 + 15.5 '+ 4.8 Hs . ? and tliis indicates, in so far as it goes, a tendency on the forehead for distances to seem longer when pressed vertically than when pressed horizontally. Of course the sum totals, which foot up the average columns, express this tendency more generally ; for instance, the final fraction, for the forehead, shows for all the categories and degrees of difference worked with, that, on the average, the vertical distances seem longer than the horizontal ones by a ratio of |-§f |, or about twice out of three times. Of course the above ratios do not express the amount of error by which the distance is over judged, or fore- shortened. Some idea of this amount may, however, be gained by casting the eye along the line of fractions from left to right, while observing the amount of differ- ence between the compared distances as indicated in the headings of the several columns. Thus it is easily seen that this difference for the first column is 1 millimeter, for the second column 2 millimeters, the next 3, the next 4, and the next 5, thus increasing from left to right through those columns wherein the standard distance is always 1 centimeter. The ratios of the first line of fractions should increase, and those of the second line OUE NOTIONS OF NUMBER AND SPACE. 67 decrease from left to right .through each set of 5 columns. What this expresses, in a general way, under the Psychophysic Law, as to the amount of error by which vertical distances seem longer than horizontal ones, is obvious, though in itself the amount cannot, from these figures, be exactly determined. i. 68 OUli NOTIONS OF NUMBEH AND SPACE. Table 28. Experiment F. — Comparing Vertical Distances. ^^ S •s ft o 1 1 1 1 1 2 2 Horizontal Distances. .9 .8 .7 .6 .5 1.9 1.8 T^ 19.25 15.75 13.25 14.0 12.75 22.75 26.25 a H 80.75 84.25 86.75 86.0 87.25 87.25 73.75 o Oi > H 32.5 31.75 30.5 39.5 46.75 35.25 44.5 <1 V 67.5 68.25 69.5 61.5 53.25 64.75 55.5 g 03 V 46.5 41.5 30.25 25.75 20.75 66.75 64.50 H 27.75 37.75 45.75 33.0 37.0 48.25 43.75 < < V 72.25 62.25 54.25 67.0 63.0 51.75 56.25 OUR NOTIONS OF NUMBER AND SPACE. 69 horizontal and vertical distances. 2 2 2 3 3 3 3 3 Totals. Sum To- 1.7 1.6 1.5 2.9 2.8 2.7 2.G 2.5 tals. 20.75 79.25 14.25 85.75 12.25 87.75 32.0 68.0 38.5 61.5 32.75 67.25 27.5 72.5 21.0 79.0 i 323 1177 1022 45.25 54.75 64.0 36.0 61.75 38.25 49.75 50.25 43.0 57.0 56.5 43.5 60.0 40.0 58.25 41.75 699 801 1978 62.5 37.5 60.5 39.5 48.5 51.5 75.25 24.75 72.25 27.75 82.0 18.0 71.0 29.0 68.0 32.0 836 664 1812 66.5 33.5 65.0 35.0 68.0 32.0 77.0 23.0 79.25 20.75 77.25 22.75 81.25 18.75 79.25 20.75 976 524 1188 37.5 62.5 36.25 63.75 33.75 66.25 47.75 52.25 47.25 52.75 48.0 52.0 52.25 47.75 57.5 42.5 614 886 1332 39.5 60.5 49.75 50.25 43.0 57.0 55.75 44.25 02.75 37.25 62.5 37.5 64.5 35.5 66.75 33.25 718 782 1668 A STUDY OF THE RESULTS. NUMBER. § 1. The two upper blocks of figures, of Tables 1 to 7, relate to the "number judgments" obtained in Exper- iment A. Examination of these figures discloses that their distribution in every block is governed by certain main laws, all holding good throughout, thougli with variable force in their relative manifestations under the different conditions of the experiments. We have to note these laws and to inquire their meaning. The first is a law of chance, imposed by tlie methods of the experiment. We note that in the second block of figures, through all the tables, the values in the " II pin " or left-hand columns are invariably plus, and those in the right-hand or "V pin" column are always minus. The reason for this is obvious when I explain, that the subject always knew what categories were being used in the experiments.^ In the present "num- ber judgments " he knew there could never l)e less than two pins, nor more than five. Consequently there could 1 It is best to explain these to the subject from the first, as it is impossible to keep him from forming notions about them diu-ing the course of the experiments. 72 OUR NOTIONS OF NUMBER AND SPACE. never be any minus errors in the number judgments of "II pins," and never any plus errors in those of '*V pins." By mathematical calculation we could theo- retically deduct the effects of chance from our work ; but as we should then have a set of figures but little if any more significant for our purpose than those already given, I have neglected such theoretical calculations.^ Other things being equal, the less the average amount of error made in judging the pins, the greater number of times per hundred should the pins be judged cor- rectly. Consequently the distribution of the figures in the upper row of blocks ought ahvays to stand in inverse ratio to that of the corresponding figures of the blocks below. Examination will show this to be the case. § 2. If we again study the two upper blocks in our first seven tables, we shall note the second of our three laws of distribution. It may be stated as follows :, The longer the distance, the more accurate the judgments. 1 A few points, however, may well be borne in mind. Namely : that in the second row of blocks, the effect of chance is to make the values in the "II pin" and in the "III pin" columns, respect- ively, as much too great (+) as in the "IV pin" and "Vpin" columns they are too small (— ) ; and also that, respectively, the -f- and — errors of III and IV ought to be less than the like errors of II and V. If all the errors were solely due to chance, then all the values of II and III should be -|- , and all of IV and V should be — , and II and III should, respectively, balance IV and V. All deviations from such distribution must indicate the influence of other laws yet to be determined. OUn NOTIONS OF NUMBER AND SPACE. i 6 This law appears simple enough so long as we study alone the judgments made of II pins, and with- out asking whi/ any such law ought to hold good. Practically, throughout all the blocks and tables, the '*II pin" columns show regularly increasing accu- racy as the distance increases from 1 to 3 cm.; that is, downward in these columns. Moreover, the "III pin " columns show openly, in general, a tendency to follow this same law. We may note, however, that usually, throughout the shorter distances, these "III pin " judgments incline to follow an opposite course ; beginning at the top of these columns the accuracy of judgment appears to decrease, till a certain length of distance is reached (differing according to the region of body studied), whence onward, with increasing distance, the figures follow the law at present in hand as regu- larly as do the judgments of II pins throughout. Al- ready this regularity of exception to our present law foreshadows the cooperation of a third law. But we feel much more the need of some such further principle to account for our results, as we come to examine columns IV and V. Here we appear to have, com- pletely reversed, the very common laAv of experience, which certainly holds good for two pins, and if we accepted the mere empirical evidence of these figures, we should conclude that, in judging four and five piris, the effect of spreading them further apart decreased 74 OUE NOTIONS OF NUMBER AND SPACE. the accuracy of our judgments. This brings us to our Third Law. § 3. If we say that, with decreasing distance, uncer- tainty of judgment increases, we shall merely be stating our old Law Two in a new form. But ''increasing uncertainty" means increasing liability to spread our judgments over categories other than the correct one ; it means increasing tendency for the mind to pitch its judgments upon its possible categories, rather than upon the right particular one ; to depart from highly specialized, accurate, and fixed habit, to less definite, and less rigidly developed habit. If now, according to Law Two, with decreasing dis- tance we have inci'easing uncertainty, then, by Law Three, with decreasing distance we have a tendency to spread the increasing uncertainty more and more over the wider field of possible judgments. By '' possible " Ave must not mean, as limited to the four number categories of pins in our experiment, but possible by our whole nature. The whole range of numerical categories, which is possible in this larger sense, is very great, and the relaxation of accuracy or of particular habits does not take place in equal pro- portion toward all. For the four number categories used in our experiments, the uncertain judgments con- tinually drift toward higher numerical categories. The lower the numerical category the stronger is this tend- OUK NOTIONS OF NUMBER AND SPACE. 75 ency. Or at least, the drift of errors being in a general direction, we find as the outer impression is moved in that direction, the less becomes the error which is due to the drift of uncertainty. It is probable that if the numerical categories of our experiment extended high enough, we should find a place where there Avould be no tendency for uncertain judgments to drift in any single direction. The reason for all this we shall come to presently, but we may now throw our Third Law into its empirical form as follows : The lower the numerical category, the stronger is the tendency of the uncertain judgments to drift toward overestimation. § 4. Having found our three laws we will now ex- amine our tables. We shall need, here, to follow in detail but a single example, and simply because they are longer than some of the others, we will take Blocks 65 and 70 in Table 4. Beginning in the upper left- hand corner of 65 we find that, for the region here studied, two pins, when separated by a distance of 1 cm., are judged correctly only seven times out of a hundred. From the figure 7 vertically downward, the numbers increase pretty regularly till at 5 cm. two pins are judged correctly seventy-nine times. All this plainly, by reason of Law Two. Going back to the same figure 7, we see the numbers increasing hori- zontally to the right, till, with the shortest distance 76 OUR NOTIONS OF NUMBER AND SPACE. category of 1 cm. remaining constant, we find V pins judged correctly fifty -five times. All this, however, as I believe, wholly from the ''drift of error," and not as in column II through increased functional accuracy of judgment. Merely, here, Law Three overbalances Law Two, As we run down the left-hand column V, we see the number decrease pretty regularly from 55 to 28. This indicates no contradiction or suspension of Law Two, but by Law Three, with increasing distance should go decreasing drift of error; that is, less tendency of the uncertain judgments toward overestimation. If we look below, in Block 70, at the corresponding figures to the last above-mentioned, it would seem at first sight absurdly incorrect to assert overestimation for these judgments at all, for all the amounts of error are given as minus. But plainly this is explained by Law One. By the terms of the experiment there could be no judgment greater than five pins. But the "drift of error," by Law Three, was active all the time, even as against these terms, and made the values appearing here as relatively minus, really greater ; that is, less minus than they otherwise would have been. As evi- dence that the natural drift of error for the whole range of numerical judgments from II to V inclusive is positively upward, it is to be noted that the average error for all the distances, if calculated, would be plus ; that the average for all distances is + 11.44 ; and that OUR NOTIONS OF NUMBER AND SPACK. 77 a roughly estimated (leduction of the "chance" values from the various coiumns, as suggested in our discussion of Law One, would leave an unmistakable indication that the real tendency in this range is throughout toward overestimation. The blocks above examined are typical, and Ave may observe of the number judgments throughoiit our tables, that in column II the obviously controlling influence is Law Two : with increasing distance goes increasing accuracy. In column V the obvious influence is Law Three : with increasing distance goes decreas- ing uncertainty ; therefore decreasing drift toward over- estimation; therefore decreasing correction of functional inaccuracy by "local drift"; therefore fewer correct judg- ments — the influence of Law Two outweighing that of Law Three, the latter remaining active all the while. In column IV, as it should be. Law Three is less powerful than in V, but remains the obvious influence. In column III, as it should be. Law Three is strongest in the shorter distances, and with decreasing force is dominant up to a distance (about 2.5 cm. in the figures for the abdomen) when its influence gives way to that of Law Two, which holds sway thence upward. The influence of Law One we have already made sufficiently obvious. § 5. Having enabled the reader to study our tables for himself, I shall now dare to ask him to consider 78 oun NOTIONS of number and space. their content and their laws in the light of a funda- mental hypothesis as to the fornitition of numerical judgments in general.^ Attacking our Law Tavo, to discover why with in- creasing distance there should go increasing accuracy of judgment, we must first ask why the simultaneous stimulation of two points of skin lying under two pin- points, situate a proper distance apart, should ever give rise to the conception of duality at all. The reason for this is somewhat as follows. In the first place these two areas must previoush^, sometime in life, have been stimulated the one after the other in immediate succession. The conception of duality in general is a particular mental state or process which is the result of such a shock ; it is the feeling of such a shock. As such it falls under all the laws of Associa- tion and of Memory which govern other conceptions, notions, mental states, and processes. As such, regain, it is subject to all the eifects of habit which, going back a step further, govern these above laws ; and in turn the particular habits, which control the associative and perceptive activities of any conception of duality for any particular pair of points or areas in an}^ given region, depend still more fundamentally upon 1 I find the main thoughts of this hypothesis best stated in Professor James' Principles of Psychology (vol. I, chap, xi, in particular, pages 487, 488, 498). OUR NOTIONS OF NUMBER AND SPACE. 79 the average run of experience common between these two points or areas. The final habit is the resultant average of all the past liabits. On the whole, during life these particular areas have not been as frequently stimulated simultaneously as successively; consequently the successive '-'number mode" which is the conception of duality, has become more strongly established as between these two points, than has the mode native to simultaneous stimulations, namely, the conception of unity. We could, perhaps, by refined means, stimulate the total nerve ends of even these two pin-point areas in some plural form of succession. That is, we could divide them into three, four, or any number of separate groups and then stimulate these groups in succession. And if such a practice prevailed through life above all other modes of stimulating these areas collectively, then by our hypothesis, the simultaneous pressure of two pins upon these separate areas would give us the numerical conception corresponding to that mode of succession, rather than as now to the dual mode. AVhy, therefore, the pressure of two pins on separate areas commonly give us a conception of their duality, is not so much that the particular tools of stimulation then and there used are two pins, as that between those precise areas taken collectively the mode of stimulation, which, on the whole, through life has prevailed and set up its particular habit of mental reaction, has been the mode of dual succession above all others. 80 OUR NOTIONS OF NUMBER AND SPACE. This being so, the explanation of our Law Two (that with increasing distance the habits of plural numerical judgments become more accurate) is easily reached. In very small areas the nerve ends collectively are more frequently stimulated together than separately. The habit of unity prevails over all other numerical habits. Hence stimulation of such areas is most likely to give rise to the numerical conception of "one thing." This will hold good even though the tools of stimulation be the same two pins which, when set further apart, will invariably give rise to the notion of their being two ; if the two pins be set too near together they will ordi- narily be judged as "one." When, now, we come to spread this spacing toward wider distances, Ave depart from conditions where the unitary habit is strong toward those where this habit is less strong, and where the habit of duality begins to be its rival. As we go on widening we find the former continually weakening till it fails entirely, and the latter growing more and more strong till its judgments approximate absolute certainty and accuracy. Our Law Two, therefore, in so far as it relates to our judgments of two pins, is but an expression of the fundamental fact that the further two points are separated on the skin, the more confirmed become the resultant habits of experience, relative to those points, in favor of the dual mode of reaction over and above all other modes of numerical reaction. OUR NOTIONS OF NUMBER AND SPACE. 81 The law holds equally good and expresses precisely similar facts in higher numerical judgments. Judg- ments of "three," and of ''four/' are particular mental states or conceptions, based upon series of three, and of four cuccessive sense impressions, in a manner strictly analogous to that in which judgments of "two" are formed. Other things being equal, the further apart any x number of separate points or areas of skin shall be, the more through life are all those stimulations or impressions, which bring all those points into collective relationship with one another, likely to fall into a series of X successive impressions rather than into any other particular combination collectively of those several points. The influence of this truth upon our mental habits works as a factor in the formation of the judg- ments of one numerical category as certainly as in the formation of those of another. But while it may be an unmixed influence in one category — such as we discov- ered it to be in the "II Pin" columns of our tables — it may be mixed with other influences in judgments of other categories, such as, for instance, those of columns IV and Y. It remains to glance at some of these other influences, in the light of our general hypothesis. § 6. One of these is the influence, under Law Three, by reason of which the drift of errors in uncertain judgments (at least in those for the categories used in our experiments) is constantly toward overestimation. 82 OUR NOTIONS OF NUMBER AND SPACE. We have pointed out that in departing from liabitual reaction in a single way, we sink to a looser and wider range of reactions. If the bond of connection or of associative habit for the uncertain sense impressions were equally strong toward all the possible numerical reactions, then it would be easy to see why, for judg- ments of the lower numbers, the average drift of error would constantly be toward over-estimation. It would be a mere matter of mathematical chance, or, as we might say here, of psychological chance. The drift being equal, the average drift for the lower numbers would necessarily be upward. How far this serves to explain the actual drift of the errors of uncertainty in our experiments, or whether it is a correct explanation at all, I cannot at present with any certainty decide. Personally, however, I incline to look upon this drift as the expression of a loose and inexact general habit of the whole brain, or of a large sphere of it, to act most strongly in the direction of the general average of the entire range of experiences embraced in that general habit ; this rather than to conceive of the many errors, actually committed in uncertainty, as so many accidental reactions in several loosely incitable but definitely directed habits. § 7. Further evidence for our hypothesis appears when Ave compare the results obtained upon one region of the body with those of another, but for reasons that OUR NOTIONS OF NtJ^NIBER AND SPACE. 83 will become evident I will postpone considering this matter. § 8. With reference to Tables 5 and 6, both refer to the same region of skin — the forearm. The difference between the method pursued in the regular experiments upon the forearm (Table 3) and that which gave us Table o is, that in the former the , row of pins was rocked lengthwise upon the skin, and in the latter all the pins were pressed on evenly and at once. The former gave series of impressions precisely like those which originally gave rise to our numerical judgments. They are such original impressions as our habits of numerical judgment at first hand are founded upon. The latter method gave us only simultaneous impres- sions ; these were no longer impressions like the original impressions ; were not the old successions happening over again ; and the judgments Avhich followed were only weakened imitations of former judgments awakened at second-hand through memory and association. Xow, since these latter judgments depend more upon memory than do the former, they must be more uncertain tlian judgments based upon successive impressions. But with increased uncertainty- should go more marked exhibition of Law Three ; and if in our tables we discover this, we should count it as confirmatory of our general hypothesis as outlined from the beginning of our paper. '■-' \ 84 OUR NOTIONS OF NUMBER AND SPACE. Turning to Block 83 of Table 5 we do observe just such an increased influence of Law Three in proportion to the relative influence of Law Two, as we have spoken of. Comparing these figures Avith the corresponding ones for our " regular " experiment on the forearm (Block 45, Table 3), we find unmistakable evidence of greater uncertainty and of the distribution of the conse- quent errors according to the law which we have laid down and given the reasons for. In column II, where the effect of Law Two always is most obvious, we see the accuracy of judgment reduced by some 30 to 50 per cent., while in column V, where Law Three is most evident, we find an increase of its influence about the same in amount. Moreover, as the method of applying the pins was changed alike for the whole scale of distances, so the results show a tolerably equal amount of change throughout the block ; that is, the increased drift toward over -estimation is a pretty equal one throughout. All the numbers in column V are approxi- mately as much increased as those in column II are decreased. In short, throughout, with a like change of method we see a like change due to Law Three. § 9. For the results shown in Table 6, not only were the pins applied evenly without rocking but, by the effort of will, the attention was confined strictly to the number of pin-points clearly felt. Practically, this means that the mind was not permitted to range up and OUR NOTIONS OF NU:*LBER AND SPACE. 85 down the line of pins, as it sat on the skin, "listening" here and there as to whether a pin really was felt then or not, and, by a continuation of this process, coupled with a knowledge which the subject always had of all the categories which were being used, to reckon out just what combination of pins and distance was at the moment being applied. I say the mind was not per- mitted to range up and down. This is but saying that the fundamental process Avhich is the basis of Law Three was here not permitted to play. The memory process was shut out, and consequently the drift toward over-estimation was shut out. Xot only this, but I am inclined to think that the actual results reveal to us, in a strikingly significant manner, the working of memory in reviving through simultaneous impressions the numerical judgments which originally, at least, were the effects of successive impressions. The " ranging up and down of the mind " in search of the proper category is much like actually playing over again in imaginary processes, the actual successions of the original events ; and the effects of inhibiting this '■ ranging," as shown in Table 6, are so surprising as to emphasize the question as to whether or not such imaginary successions, in some form, per- haps almost infinitely compressed, are not what really happen, in the formation of all numerical judgment from a simultaneous impression, and whether, therefore, 86 OUR NOTIONS OF NUMBER AND SPACE. they are not absolutely necessary to the formation of such judgments ? Under the influence of the inhibition of such processes, column V of Table G shows, in some places, an absolute lack of correct judgments of the higher categories — those which would require the greatest amount of this imaginary play — and shows but a very small number of such judgments anywhere throughout the column. iVTothing I have said, however, must be mistaken for a premature inclination to decide this matter.^ DISTANCE. § 10. When we draw a pencil-point along the skin we stimulate successively the nerve ends lying in the line drawn. In essential nature such an event is no different from those primitive occurrences which, accord- ing to our hypothesis, give origin to our number judg- ments. They are both based on serial impressions. The difference between the number judgments and the distance judgments lies chiefly in the nature of the successions which characterize each. In the former the terms of the series are comparatively few, the suc- cessions are sharply marked-off, and slow ; so slow and marked that we note and count them — successively give 1 1 am sure that our experiments here are capable of teaching us an important lesson as to the intimate nature of the processes of attention in the formation of judgments in general. OUK ]S'OT10^'S OF ^-U:MBER AND SPACE. 87 names to them. Thus : one — two ; or one — two — three. When a line is drawn along the skin a myriad of nerve ends are stimulated in relatively rapid and unmarked successions ; so rapid and unbroken that we do not note the separate terms, nor individually count them. The difference is that in one case we say, " one — two — three," and in tlie other, '' so ma-a-a-a-any '' or '• so fa-a-a-a-a-a-a-ar." Each specific length, however, of the distance series has a nature of its own which is the basis of each one of our specific categories of distance judgment (our ideas of particular distance), in the same Avay tliat the number series each have a particular length or number of terms which is the basis of each category of number judgments. § 11. We have said that, given a definite set or arrangement of nerve ends, until this set, some time in life, be broken up and its parts be first stimulated in some sort of succession, any sort of simultaneous stimu- lation of this particular set will not arouse any sort of plural category whatever. AVe now say that it will arouse no conception of distance whatever. Any set of . nerve ends which has never been stimulated except in complete simultaneity wall give us the same sort of mental response or experience when distributed over the surface of the skin in a compact bunch, as when distributed in anv sort of lineal arrangement. 88 (3UR NOTIONS OF NUMBER AND SPACE. § 12. Yet it is the fixed lineal or spatial arrange- ment of our dermal nerve ends that determines our particular distance and space conceptions regarding them. This happens because it is the fixedness of each particular arrangement that determines what manner of serial stimulation through life shall most frequently fall to the lot of that collective group of nerve ends. If two nerve ends are permanently located immediately beside each other they are likely to be stimulated more times during life simnltaneonsly than successively ; and consequently, upon simultaneous stimulation, are more likely not to arouse any notion of distance betv/een these two points than to arouse such. If two nerves are fixed widely apart they are more likely than otherwise, in the whole of life, to be affected by various moving impressions which shall first stimiilate one nerve at a given point in the series of impressions and the other nerve at another point in the series. Consequently it is more likely than not that some sort of distance cate- gory will be developed and attached to the collective stimulation of these two separate points. § 13. Every pair of separate points is likely to be stimulated by all sorts of lineal impressions moving first through one point and then through the other. This, in the same way that it is possible to draw all sorts of lines (straight, broken, and curved) through any two points of our skin. The question then arises how OUR NOTIONS OF NUMBER AND SPACE. 89 any particular distance category, corresponding to some particular length of moving series, comes to be so joined to any two particular points that we commonly judge them to be a definite, actual distance apart? This is not difficult to answer when we recall that the kind of memory category that is awakened by the simultaneous stimulation of any definite combination of nerve ends, is based on the dominant and average habit which is the resultant of the experiences which have most frequently combined that particular collection of nerve ends. With reference to the skin, it is evident that right-line movements are likely to prevail between points separated by a few centimeters (as in our experi- ments) far and away above any other particular form of lineal movement. § 14. But the rate of drawing a line may be infi- nitely variable, and the time element of the series is the most important thing of all, according to our hypothesis, as the primitive basis of distance measurement. Here, again, we see that the single definite habit, which finally results froin the particular modifications of each one of the infinite number of infinitely varied time series expe- rienced through life between every pair of dermal points, solves the difficulty. The average of such an infinity of time experiences for any pair of points would, other things being equal, be proportional to the actual, fixed right-line distance between the points. Consequently, 90 OUE, NOTIONS OF NUMBER AND SPACE. iu proportion as the resultant or prevailing liabit is a fixed and accurate one does its mental judgment or perception accurately represent or correspond to the actual right-line distance. § l.j. At this point many difficulties arise if ^ve inquire as to the intimate and specific nature of our different distance perceptions. All that we have said supposes them to be based upon reawakened time series of correspondingly specific length or nature. Yet we surely must reject the notion that our hasty judgments of different distances are always of the same absolute and specific time lengths, all in due proportions. But if not, how do they preserve any sort of proportions between themselves, duly representative of actual outer differences ? This leads us to one of the most obscure regions of psychology. To me the following hypothesis seems both more clear and more justifiable than the average psycho- logical hypothesis of modern text books relative to the intrinsic nature of a judgment. It is the connective or associative function of any mental processes or habit that is of importance in the formation of accurate thoughts and judgments, rather than the nature of its content. If the function is accurate and specific, the judgment will be accurate and specific. The function of the specific perception or judgment is, to make the proper connection between the outer event or impres- OUR NOTIONS OF NUMBER AND STACK. 01 sion and certain following thoughts, perceptions, or associations about that event or impression. If the proper specilic connection is made, it makes no, or little, difference to the accuracy of the thinking, what the specific nature of the mental content of the judgmental in itself may be, either qualitatively or in absolute time- duration. Suppose the perceptive connection is to be with the motor idea which incites us to say, " one centimeter.'"' The connective activity may occupy an absolutely longer or shorter time, and this make no difference provided the nature of the process is such that the proper motor-idea is eventually incited. This being so, I think we may easily conceive how that, though our various specific judgments are all based upon habits which are the resultants and the correspondents of time series of relatively different absolute lengths, they yet may not themselves occupy absolute intervals all proportionally different, nor their conscious content be of any given specific phenomenal nature. Indeed we may easily conceive how the really decisive and dis- tinctive link should be wholly an unconscious mental process. If we take our stand on the Summation Theory, we can conceive how the specific intensive force or constituency of the differently summated series might determine the proper connection independently of specific time dvirations. Or we may with plausibility conceive of some specific anatomical arrangement of 92 OUR NOTIONS OF NUMBER AND SPACE. brain parts, correspondent to each specific judgment, whose liability to be affected as a whole, depended upon specific series of stimulations of absolute length, and yet whose subsequent activity as specific wholes in memory should not necessarily occupy always the same absolute time-rhythms. Or, better still, perhaps we may conceive of a combination of intensive, anatomical, and spatial distributions which shall mediate the activities correspondent to the correct specific judgments and make accurate connection thereby with the proper associative thoughts, yet do this quite independently of an absolute time performance which should be pre- served constant for every repetition of the judgment. § 16. With so much of our hypothesis before us we may now come nearer the laws which specially govern our tables. To begin with, we may note that in the main all we have said about ''chance" must hold good as much for the two lower rows of blocks in our tables — those which relate to the distance judgments — as it does for the two upper rows, which relate to the number judgments. § 17. According to Law Two the numerical judg- ments were the more accurate, the greater was the distance. Under what we may call the same law, we sliall now find the accuracy of the distance judg- ments also, as a rule, increasing with tlie actual distance. But the reasons for this law are neither OUR NOTIONS OF NUMBEU AND SPACE. 93 SO plain nor so simple as previously. Since our percep- tion of the distance between two points of skin is based, primitively, upon the predominance of movements in the right-line joining those points over moverdents in any other particular line or combination of lines ; and since the greater the distance the less should be this predominance, therefore, were these the only principles determining Law Two for distance, we ought to find increasing inaccuracy with increasing distance. It is probable that this principle has its proper influ- ence, primitively, in developing our notions of particular distances everywhere, and very likely there are lengths of distance for certain stretches or regions of skin where the, principle is a predominating one; but within the short distances and regions investigated by us, other principles come in which counteract and far outweigh it. § IS. One of tliese other principles is similar to the one which explained Law Two under " Number." We may describe it as follows : Our memories concerning the distance between two points are based on move- ments between these points ; the longer the distance, the more are the two points disassociated by these move- ments; that is, the more do these movement-habits weigh against the tendency for the two points to fuse spatially in memory ; they tend so to fuse in proportion as the points are habitually combined simultaneously. There- 94 OUR NOTIONS OF NUJNIBER AND SrACE. fore, in forming the prevailing memory-habit, tlie disas- sociation of movement is always opposed to the influence of simultaneous combination. Thus Law Two in dis- tance, or at least within short distances, formulates, as in number, the increasing tendency of the influences of the successive experiences of life to prevail w^th increasing actual distance over the influence of the simultaneous experiences of life. Put into every-day language, this is but saying that Ave are more likely to judge the distance between two points accurately, the more capa- ble we are of forming a distinct conception of their sej^arateness. Put more technically: the further apart two points of skin are, not only is our conception of their ^eparateness heightened through disassociation due to the movements between those single points, but also it is heightened through the growing differentiation in the grouping of the local associations around each point. If the di^stance increase enough, the two points w^ill habitually fall into strikingly incongruous groups. Por instance, the point of the toe and the back of the head. Now, it is plain to any one having much experience as a subject in the experiment we are discussing, that our judgments are not confined to data based on the precise line join- ing the terminal pins. Rather, we form a notion of the separate and particular region w^here we find one pin to be, then another notion of the region of the other pin, OUR NOTIONS OF NUMBER AND SPACE. 95 and we then say that one region is '' so far " from the other. This being so, we see at once that the particuhir judgment of distance which we finally render, is not based on the simple disassociation-force of the single line between the two pins, but by the whole disassocia- tion-force of the two "region groupings" — the whole force of the region's associations to fall into distinct and separate groupings rather than to fuse into an unseparated single spot. § 19. Still another principle works in favor of Law Two. Our voluntary measurements of shortest-distance are always in straight lines. When we train our judg- ments of distances, we are always training our right-line memories. We know how great are the results of train- ing, and how great must be the influence through life of joining the right-line memory to, and weaving it into, the concept of measuring. § 20. I think it should now be clear how increasing distance works to increase the accuracy of estimating the distance between points. The greater the distance, the sharper and stronger will be our conception of the separateness of the two regions. The sharper and stronger this conception of the regions, the more highly developed in connection therewith will be the memory effects of the voluntary measuring of the points. The voluntary measurements will all be based on right-line experiences. The three concepts — of the regions, of 96 OUR NOTIONS OF NUMBER AND SPACE. the measuring, and of the right-line — will fuse, and act as a whole ; the resulting judgment will be strong, clear, and accurate, in proportion to the strength, clear- ness, and accuracy of the united conceptions. As two of the conceptions are increasingly sharp and clear with increasing distance, so does the total resultant judgment increase in accuracy with increasing distance. It remains to be said of Law Two, that for very short, or sub-threshold distances, we should expect to find its effects very weak ; for them all the elements of the law would be very poorly developed. The conception of the separateness of the points would be faint; the skill of measuring such unusual distances would be scanty; and the immediate impressions would tend to group into "spots" rather than to revive the serial memories of right-lines. The judgments would be uncertain, and would, consequently, fall under the influence of the laws of uncertainty. This brings us to the law of uncertainty, which we called Law Three. § 21, Law Three, in the number-judgments, formu- lated the drift of the errors of uncertainty. It will do the same here. As before, the drift will be toward the average of the possible categories. But the spatial categories are more variously conditioned for different regions of the skin, than are the distance-categories for the same regions. That is, the character of the spatial experiences depend far more on the shape, area, and OUR NOTIONS OP NUIVIBER AND SPACE. 97 contour of a particular area of skin, or member of the body, than do the numerical experiences. For instance, the distance series native to the tip of the tongue would never be long, while the number series might run as high there as anywhere. We shall then expect to find over-estimates and under-estimates in any fixed scale of categories of distance, like those of our tables, to be greatly variable between different regions of skin. While we shall find ourselves obliged to examine each region more carefully by itself, to determine the precise influence of Law Two in each case, yet it will be just this lawful variableness which will be of signifi- cance to us when, as we propose, we come to test the truth of the Genetic Hypothesis, as a Avhole, by com- parative studies. • § 22. Beside the above laws, we shall discover for distance-judgments still another law, which played no part in number-judgments. We Avill call it Law Four, and it may be stated as follows, i.e., The greater the number of pins, the shorter will be the estimated dis- tance. And since the right-line distance will be both the shortest distance and the correct distance, we may say that, other things being equal, by Law Four: The greater the number of pins, the more accurate should be the distance-judgment. This is to be accounted for as follows : Our judgments are based vipon the memory habits joined to particular 98 OUR NOTIONS OF NUMBER AND SPACE. nerve-ends. Also, our distance-judgments are based upon right-line movements between points. Now it is plain tliat, in these original movements, every nerve lying in the line of any movement would be as much joined to the resultant memor}^ effects of that move- ment, other things being equal, as would any other nerve lying in that line. Consequently, although we shall discover other reasons than the above why that particular distance-memory, as a whole, becomes more joined to the end-points of the line than to intermediate points, Ave still may see from the above reasons why each intermediate point is very intimately joined with that particular memory. This being so, it is easy to see Avhy every additional pin introduced between the two end pins in our line of pins should be an additional stimulant to the revival of the proper perception and judgment. Each pin in the right line is a guide toward the distance perception, being based on the right-line memory, ratlier than on the possible memory of innu- merable other lines. Consequently, the greater the number of pins in our experiment, the more accurate should the judgments of distance be.^ iThe reason wliy some other category of distance rises upon simultaneous stimulation of two intermediate points of an orig- inal right-line movement (namely, that of the shorter distance between the intermediate points, rather than the original cate- goiry) is plainly not because the longer category has no strength, but because the shorter category has the stronger and prevailing OUR NOTIONS OF NUMBER AND SPACE. 99 § 23. Having explained our laws we will now exam- ine our tables. Turning for illustration to Blocks 75 and 80 of Table 4, which contain the figures corresponding to those already used in illustrating the laws of numeri- cal judgments, we first observe in Block 80 that all the values in the upper horizontal line are plus, and all of those in the lowest horizontal line (above the averages) are minus. This is the effect of "chance," and Law One, which, it will be observed, now works in even horizontal lines, up and down, from top to bottom of the block, instead of right and left, as before, in the number-judgments. It would be corrected by sub- tracting a maximum amount from the (+) values in the top line, and adding a like amount to the (— ) values in the bottom line, and grading proportional corrections strength, as between these two points. And the reason why the several categories, corresponding to each distance between each intermediate point of pins, does not rise to perception, under simultaneous impression of the whole line of pins, is not because there is no tendency for the several shorter categories to rise, but because, again, the tendency for the single longer category is the prevailing category. Why the longer one should be stronger than the shorter ones is plain, if we remember that each intermediate pin would have some tendency to call up the outside category, while the pins outside of each intermediate pair would not have equal tendency to call up the intermediate category. Why we can think alone in the one strongest category, and cannot think in all the categories at the same time, lies, very probably, somewhat in the fact that the same brain parts are likely to be demanded simultaneously in the several categories, and can act only iflu^^JU^^^ one line of strongest tendency. V-tfe -i <^l \ <<* 100 OUR NOTIONS OF NUMBEK AND SPACE. from these extreme categories toward the middle cate- gory of 3 cm., where the influence of the law is negative and zero. § 24. We next observe the effects of Law Two. By this, all the judgments of distances above the threshold distances should increase in accuracy with increase of distance. Assuming the threshold for this region to be about 3.5 cm., we see the number of correct judg- ments increasing regularly Avitli increase of distance throughout the remainder of the blocks. The average in Block 75, for 3.5 cm. is 20.5; for 5 cm. is 40.2. That the laAv begins to have effect even in the short category of 1.5 cm. is obvious from tlie figvires. § 25. Law Tliree shows a slight tendency toward over-estimation in the region of the abdomen throughout. The total average in Block 80 is +3.46, and the average for the middle category of 3 cm. is -f" 5.6. The tend- ency is, however, so light, that under the shortening influence of Law Four, the actual tendency is toward under-estimation for V pins, even in the short distances, where by effect of Law One, the actual judgments should show plus values. § 26. The effects of Law Four are most obvious in the 1-cm. judgments, and in the horizontal averages of Block 75, and they show markedly in Block 80 through- out. The effects are exhibited as increasing accuracy from left to right, horizontally, across the four columns OUR NOTIONS OF NU3IBEU AND SPACE. 101 of pins. Thus in Block 75, we read for the 1-cm. judg- ments, 42, 53, 66, 78 ; and for the averages, 30.6, 30.8, 31.3, 31.6; and the decrease of the amounts of error in Block 80 may be illustrated by the averages, which read + 10.3, + G-O, + •^, - 2.6. § 27. To test the united influence of our various laws we will now examine Blocks 75 and 80 more inti- mately. Studying the top line of Block 80, we see as folloAvs : Law One makes all tlie values more plus than actually they should be. Law Two shows a maximum of uncertainty for this, the shortest category of distance ; the average error, + 26.0, would be the greatest for any of the distances, even after making corrections for Law One. Law Three shows a drift of error proportionate both to the uncertainties due to Law Two, and to the uncertainties due to Law Four. Proportionate to Law Two, there is over-estimation holding good for the aver- age of all the minimal distance-judgments, the average error being -{- 26.0, — a sum that indicates over-esti- mation after correction for Law One. Proportionate to Law Four the drift of error is greatest where the uncertainty by Law Four is greatest, i.e., where the pins are fewest, and decreasing as the pins increase from left to right ; thus -f 37.0, + 29.8, + 21.0, + 16.1. Law Four shows a shortening of the judgments with increase of pins ; this is shown in the four numbers last quoted. Of course the top line of Block 75 would 102 OUR NOTIONS OF NUMBER AND SPACE. show effects correspouding to the above, if correspond- ingly analyzed. Next, examining the figures of column V in both blocks, we find : Law One making the judgments too long in the short distances and too short in the long distances. Law Two makes the judgments more accu- rate with increase of distance. This is obvious in the lower half of Block 75, where the number of correct judgments increases with the tolerable regularity of 20, 23, 20, 39 for the distances 3.5-5 cm. The real effects corresponding to these, in Block 80, are obscured by reason of the fact that the minus quantities due to Law One increase in this column from the middle down- ward, at a ratio greater than that by which the minus values, resulting from drift of error (under-estimation in this V-pin column) decrease through increase of accu- racy due to Law Two. Make correction of Law One, and Law Two is very evident. The real effects of Law Two in the upper halves of the two blocks is obscured in a like manner. That is, in the upper half of Block 80, we have minus values even against the influence of Law One, which theoretically should give stronger plus values ; consequently, here there must be under-estima- tion by Law Three, heightened through the influence of the high number of pins, i.e., by Law Four. Now, since the plus influence of Law One falls off, with increasing distance, faster than do the united minus OUR NOTIONS OB^ NUMBER AND SPACE. 103 values of Laws Three and Four, we therefore have the apparent contradiction of Law Two ; the contradiction, however, being only apparent, and due, as we have seen, to the compensating influences of the other laws. The influences of Laws Three and Four are, from the fore- going, sufficiently plain, and the entire distribution of the figures in column Y should now also be clear, as displaying and agreeing with the united influences of our several laws. Examining the lower horizontal line (above the aver- ages) to test our laws in the higher distances, we find as follows : Law One gives minus errors throughout. Law Two gives maximum accuracy for the longest distance ; as is evident from the large number of correct judg- ments shown in Block 75, and the small average error that would remain after correcting the minus-constant of Law One. Law Three averaged for the four columns of pins, shows a slight tendency to over-estimation at the distance of 5 cm.. Block 80. Law Four makes this over-estimation more pronounced in tlie low-number colvimns, and shortens it perhaps to actual under-esti- mation in the right-hand or •• V-pin " column. As the result of the combined influence of the four laws, the effects of Law Four are, in Block 75, apparently contra- dicted, that is, the actual number of correct judgments decrease from left to right (48, 44, 40, 39); but it will be easily understood from the foregoing that this 104 OUK NOTIONS OF NUMBEll AND SPACE. appearance is but the effects of the various compensa- tions which we have last above described and illustrated from the last line of Block 75. The distribution in the three remaining vertical columns are so similar to that of column V that now, having both followed in detail tlie three sides of our blocks, and also examined the general influence of each law upon each block as a whole, I think the detail of the distribution in all the columns throughout should be perfectly plain to any one upon due examination. Our sample blocks, therefore, we find conform to oiir laws. This conformation Avill have weiglit in support of our main hypothesis, just in proportion as, with integ- rity, it may be shown to be representative of a wider con- formation extending throughout the large body of our experiments. Manifestly it would be impossible within any reasonable limits of publication, to go through a detailed examination similar to the above, explicitly demonstrating the course of our laws and of our main hypothesis throughout each of the 123 blocks of figures presenting the extensive and arduous series of investi- gations classed together in this paper as Experiment A. Still less would it be possible to extend such a demonstration throughout the 365 blocks of Experi- ments A to E. This each student, according to his interest in the matter, must do for himself. But 1 assert with the confidence of long and careful study OUR NOTiONS OF NUMBER AND SPACE. 105 tliat such an examination results in undeviating and constant accumulation of evidence of the integrity of our laws wherever they are implicated, and of the soundness everywhere of the main lines of reasoning upon which they have been founded. Where at first there may appear to be contradictions, we shall discover by closer study, and especially by the comparative studies already foreshadowed, that tliese are but the ap[)arent exceptions which, when under- stood, all the more abundantly substantiate the general truths. § 28. Before leaving the present discussion of dis- tance a few tilings remain to be saiil in connection with the special tables of Experiment A. Table 5. — Here the pins were pressed evenly without rocking. We have already said, in discussing this same table under Number, that the chief effects of the change from the method of the regular experiments to the present one, ought to be a lessening of the influence of the laws most dependent upon present peripheral exci- tation, and relatively to onliance the influence of those most dependent upon memory. Law Four is of tlie former class ; its influence, there- fore, should be lessened in proportion as the stimulating influence of each pin is lessened. Comparing Block 92 of the new Table 5, with the corresponding figures of Block 60 of the "regular" Table 3 for the same region 106 OUR NOTIONS OF NUMBER AND SPACE. — the forearm — we see that this actually happened. In Block 92 there is but little shortening of the judgments, with increased number of pins. The proper averages now read -^ 20.4, + 26.3, + 25.6, + 19.2 (indicating an irregular and slight shortening from left to right with increase of pins) as against formerly : -)- 17.7, + 13.3, +9.1, +7.6 (indicating a, marked and regular shortening). Law Three is of the class most exclusively based in memory processes ; its effect by the new method, there- fore, should be enhanced, as plainly it is. The average error is markedly greater throughout, and the general drift throws the increased error more constantly toward over-estimation. This shows so plainly in the tables that the figures need not be repeated here. A striking item of confirmation of our interpretation of the com- pensating influences of the several laws, is shown in the lower line of Block 89, where, lacking the compensating influence of Law Four as described in discussing the corresponding lines of Table 4 on page 103, the number of correct judgments no longer read decreasingly from left to right, but increasingly, as lacking the influence of Law Four they ought to read (38, 41, 40, 40, in Block 89, and 57, 40, 49, 48, in Block 60). Law Two being little affected by the new method, holds its course even more obviously than it did in the regular experiments ; and this, because lacking in a I OUR NOTIONS OF NUMBER AND SPACIC. 107 greater degree the disturbing influence of Law Four, is what properly it shoukl do. § 29. Table 7 furnishes other peculiar evidence for our laws. It records the results of practice. We should suspect a i^riori that the effects of practice would not improve all our laws of judgment equally. The law which we should most expect to be improved by the specific practice of these particular experiments is Law Two ; this would chiefly result in heightening our famil- iarity with the precise regions worked on ; which in turn would enable us to disassociate more sharply and distinctly the local percepts around each pin. Particu- larly would this apply to our percepts of the end pins, for the reason that our attention in judging distance is proportionally more bestowed upon these than upon the intermediate pins. Consequently, since improvement with reference to the end pins would mean, on the whole, improvement with reference to the longer dis- tances, we should expect that the chief consequences of practice would be increased accuracy in judging the longer distances, and more pronounced effect of Law Two throughout. Examination of Blocks 110 and llo (Table 7) discovers this to be just what took place ; the average error of the longest distance is now — 5.4 as against — 12.8 formerly for the same region — the fore- arm — in Blocks 55 and 60, Table 3; and the average number of correct judgments increases now for the dis- 108 OUE NOTIONS OF NUMBER AND SPACE. tances 1 to 3 cm. by the series of figures 29.5, 33.0, 39.5, 37.0, 78.0, as against the former series of 35.7, 44.0, 47.2, 40.2, 48.5. The specific practice with the few distances actually used, Avould have little direct influence upon the memory habits of the other distances. Consequently the pull of these "possible habits" upon the drift of uncertainty, where there yet remained uncertainty, ought to be about the same as before. The figures confirm this. Accord- ing to what we have said just above about the improve- ment in Law Two, we should expect to find the shorter distances remaining unimproved as compared with the longer distances. On the average, their judgments remain equally uncertain with their previous ones. The average error for 1 cm. was formerly + 52.4, and after practice Avas + 54.0. (Separate experiments we must not expect to agree wholly.) On the whole, therefore, these figures confirm what we should have expected as the behavior of Law Two, both as to average amount of error and the direction of its drift. The matter is again confirmed when we look at the new effects of Law Four. It was noted as we became more expert in judging the distances, that we more and more based our judgments directly on the impressions of the end pins ; there was less *' reckoning" along from pin to pin, such as is based upon rocking. In other words, the effects of the intermediate pins entered less OUR NOTIONS OF NUMBER AND SPACE. 100 and less into the judgment, and this is the same as saying that Law Four woukl have less effect than before practice. The obvious consequence of this ought to be, in the tables, that the judgments should exhibit less shortening than formerly from left to right through the columns with the increasing number of pins, and particularly this sliould be most manifest where there remained the greatest uncertainty. We may now observe that this is precisely what did happen. Not only do the footings of Block 113 show little of this shortening as compared with the footings of Block fiO (+ 18.9, + 19.2, + 12.2, + 14.8 now, and + 17.7, + 13.3, + 9.1, + 7.6 formerly^, but the top lines of the two l^locks show no shortening in the 1-cm. judgments after practice, and marked shortening before practice (+69.1, + 56.0, + 44.3, + 40.0 before, + 48.0, + 60.0, + 55.0, + 53.0 after). Similar fulfillment of lawful expectations can be easily traced in the number-judgments of the same special experiment. The whole of Table 7, therefore, again affords striking confirmation of our thesis in general. § 30. A word must be said of the special experiment reported in Tables 8 and 9. If our general discussion, and in particular that part referring to Law Four, is correct, then we ought to expect that straight-edges or full lines of cardboard, pressed upon the skin, should awaken more accurate judgments than our lines of pins. 110 OUR NOTIONS OF NUMBER AND SPACE. It ought to be a case of Law Four with the number of pins raised to infinity. Tables 8 and 9 report an exper- iment for testing this matter. In considering the results it should be borne in mind that the pins give much sharper impressions than do the card edges. Yet, notwithstanding that fact, the experiment is an interesting confirmation of our general doctrines. The average errors for the scale of distances on the forearm, Block 60, in the regular experiments ran as follows : + 52.4, +27.1, —.1, —6.9, —12.8, while the corre- sponding errors with the card edges, Table 8, were : + 31.4, + 8.9, + 2.7, — .6, — 7.8. Correspondingly for the number of correct judgments we have for the pins : 35.7, 44, 47.2, 40.2, 48.5 ; and for the cards : 53, 50, 60, 72, 83. The total averages are: ''pins," +11.9 and 43.16, as against "cards," +6.9 and 63.8. Table 9, where the cards were pressed without rocking, also yields its sliare of confirmatory evidence of La,w Four, and both Table 8 and Table 9 are full of points confirm- atory of our other laws, but we have not the space here to consider them. § 31. A glance at the summaries of Experiment A, exhibited in Table 10, shows again the integrity of our laws in a strikingly impressive manner, proportionate to the extensive field of confirmatory experimentation grouped into a single view. Eut for the present we must leave our special subject of distance to study higher spatial complications. OUR NOTIONS OF NUMBER AND SPACE. Ill NUMBER-JUDGMENTS BASED ON TWO DIMENSIONS. § 32. In the Experiments B, C and D our investiga- tions attack the psychology of two- dimensioned space. Necessarily Ave shall make but little headway with it ; our " heaps of figures " will show rather what in the future is to be done, than reach complete demonstration of any kind. Xaturally we should first inquire in this new domain, whether the laws of number and distance already dis- covered are carried over into the formation of the new and more complicated judgments. Number. § 33. We will first follow the laws of number. For these we must study Tables 11 to 17. They relate to Experiment B, which was conducted with pins arranged in triangles and squares, the distance categories remain- ing as before.^ [The experiments subsequent to B do not involve number-judgments.] ^ About these B tables in general, a preliminary word of caution is needed. By a great fault not appreciated in laying out the experiment, the highest category of pins both with the triangles and with the squares was not carried down through the shortest distances. As a consequence there is much complication in the operations of Law One. This is illustrated by any one of the blocks, for instance Block 196, Table 14. It will be noted there. 112 OUR NOTIONS OF NUMBER AND SPACE. That Laws Two and Three both hold good throughout this new set of number-judgments, is seen by slight ex- amination, but certain peculiarities, to be observed here and there in the wider course of general integrity, de- mand closer consideration. Perhaps the thing that first strikes us is the fact that the number of correct judg- ments, and particularly in the short distances, is on the average much greater than when the pins were arranged in a single line. Wliat has oi;r liypothesis to say of this ? Suppose three pins. A, B and C, to be set in the corners of an equilateral triangle of 1-cm. base. According to our foregoing discussions, our ability to perceive these to be " three " will depend upon the predominance, in the combination of these three par- ticular j)oints of skin during life's experiences, of '' three termed successions " above all other modes of eoordi- nately stimulating them. What we must now ask is whether, other things being equal, this mode of combi- nation is more likely to occur when three points are arranged in a triangle, than when in a straight line. Of course it is difficult to know Avhen the <' other things " tliat the correct judgments in the " Vl-pin " cohunn fall off greatly in the larger distances where the other category of " VII pins" has been introduced, from what they were above in the sh,orter dis- tances. Plainly this is the effect of chance and from opening a new and higher category which may possibly be judged. The effects in the averages are also somewhat disturbed. With due care, however, the results may be used comparatively without falling into grave errors. OUR NOTIONS OP NUMBER AND SPACE. 113 are sufficiently equal for just comparison ; as, for instance, with reference to the distance between the pins. In the above supposed triangle the distances average 1 cm. apart. Perhaps the straight-line category coming nearest to this in Experiment A is that where the end pins are 1.5 cm. apart, and the average distance between the three pins is 1 cm., the same as in the triangle. Assuming these two arrangements as concrete examples for our comparison, we observe that the triangular offers greater likelihood for successive stim- ulation than does the lineal arrangement. This is apparent if we consider the possible movements, in the plane of the skin, of a right line which is to be con- sidered with reference to its stimulations of any three given points in the skin. If the three points are in a straight line p, the moving line I must occupy some position with reference to /> that can be determined by the angle x between the lines. For our problem the movements of I must be computed between the values for X oi and 90°. But with a: = o, that is when I is parallel Avith p, all movements of I over the skin would never be able to stimulate the three points in any sort of succession. This state of things could never happen with the points arranged in any sort of triangle. It is easy, therefore, to show by calculation that, whether the stimulations be made by continuous movements tan- gentially across the skin, or by vertical pressure upon 114 OUll NOTIONS OF NUMBER AND SPACE. the skin in successively varying positions of the stimu- lant, the chances for the proper successive combination requisite to the development of the threefold form of numerical perception for the three points would be much greater in life under triangular than under lineal arrangement. The advantage in favor of the tri- angular arrangement is markedly extended by the very important sort of genetic differentiation arrived at through the principle of Concomitant Variations. § 34. The empirical fact that our experiments show the numerical judgments to be more accurate under triangular than under lineal arrangement of the pins would, therefore, if clearly demonstrated, be in strict accord with the theoretical demands of our general thesis ; and, having the theory more fully before us, we must now examine this demonstration in our tables more particularly theretoward. Assuming that the " Ill-pin, l.o-cm." results pf Ex- periment A may be compared with the " Ill-pin, 1-cm." results of Experiment B — an assumption which, as we will presently show, favors the right-line arrangement — we get from the several tables the following averages for number of correct judgments and for average error. The figures in the left-hand column refer to the lineal, those in the right-hand to the triangular arrangement : OUK, NOTIONS OF NUMBER AND SPACE. 115 Toncfiie Forehead Forearm Abdomen Block 5 " 10 Block 25 " 30 Block 45 " 50 Block Go " 70 Line. 99 + .o 27 + 24.8 Tkiajnole. 34 + 9.3 19 + 39.8 98 + .6 Block 128 " 133 20 + 49.3 Block 148 153 28 + 38.7 Block 173 " 178 4(; + 32.0 Block 190 199 The above figures are given without correction being made for Law One. To make such corrections w^e should have to subtract much larger amounts from the " triangle " results than from the " line " results ; we should liave to do this because of the difference in position relative to Law^ One of the " Ill-pin " column in the two cases. I will not attempt the proper correc- tions, but it will be evident to any one upon due consideration that, within very safe estimates, they would demonstrate conclusively the greater accuracy of the judgments under the triangular arrangement than under the lineal. Upon the same basis of average distances, if we com- pare the 3-cm. judgments of Experiment A with those at 2 em. in Experiment B, we get the following : 116 OUE NOTIONS OF NUMBER AND SPACE. Line. Tkiancli;. Tongue S Block 5 / " 10 99 + .3 99 + .2 Block 128 " 133 rorehead \ Block 25 1 " 30 00 + 4.3 50 + 29.8 Block 148 " 153 Forearm \ Block 45 ( " 50 27 + .9 40 + 28.0 Block 173 " 178 Abdomen ( Block 05 1 "70 28 + 29.1 50 + 24.0 Block 190 " 199 These figures also, when proper corrections sliould be made for Law One, would show the superiority of the triangle judgments to be very marked. § 35. It should now be noted that the above method of averaging the distances is very unfair in favor of the lineal arrangement. The judgments at 2 cm. ar,e better than at 1 cm., and manifestly it would not be right to average 100 of the former against 200 of the latter. But this is practically what we do in the above method when we make the 2 cm. of distance between the two end pins in the lineal arrangement offset two distances of 1 cm. each in the triangular arrangement. This should be borne in mind in making corrections in the two above tables and for the comparisons now in hand all through. OUR NOTIONS OF NUMBER AND SPACE. 117 § 36. 80 much for tliree pins. A similar compari- son with the foregoing may now be made for the "fonr- pin" judgments. For this, by the above method of simply averaging the inter-distances, we should have to compare the 1-cm. judgments of four pins in a square, with the 2.5-cm. judgments of four pins in a line, and such violent errors would arise from offsetting 2.5-cm. judgments with two and a half times as many 1-cm. judgments as to make such comparisons wholly inad- missible. AVe can, however, arrive at the desired information by a more satisfactory method. In Table 17, Block 210, we find the total average of correct judg- ments summarized for all the regions of skin worked on, to be: for ''III pins," 48.95, and for ''IV pins," 54.60. In Block 217, the corresponding average error for ''III pins" is +23.2; for "IV pins," +17.0. As the conditions of Law One were precisely similar for these two sets of figures, they show conclusively that four ])ins in a square are more accurately judged tlian three q)ins in a triangle. We have already demonstrated in our experiments that three pins in a triangle are judged better than three in a line. To complete our proof, tlierefore, that four in a square are judged more accurately than four in a line, we have but to show that three in a line are better judged than four in a line. It is true that the general summary of the line experi- ments, Table 10, Block 118, shows 49.2 correct judg- 118 OUR NOTIONS OF NUMBER AND SPACE. ments for "IV pins," and only 37.7 for "III pins." But before we take this as evidence tliat four pins are better judged than three, we must again make the proper corrections. We must remember that all through Experiment A we found over-estimation ; that the posi- tion of the "IV-pin" column under Law One apparently offset this general over-estimation, making the "IV-pin" judgments appear unduly accurate, Avhile the position of the "Ill-pin" column augmented the over-estimation, and heightened the errors. The figures as given, there- fore, show an illusive comparison. The illusion is greatest where there is the greatest over-estimation, namely, in the shorter distances. It will be observed that the longer distances throughout give undoubted evidence, even without proper corrections, that three- pins are fundamentally judged with greater accuracy than four, the arrangements being the same. But mak- ing the proper allowances for the compensations between Law One and Law Two, and the fact which common sense would assert from the outset, that in similar lineal arrangement three pins are judged more easily than four, will, I think, receive from the figures of Experi- ment A throughout most unmistakable demonstration. This being so, it is therewith also demonstrated that four pins in a square are judged better than four pins in a line, which was the main proposition under consideration. OUR NOTIONS OF NUMBER AND SPACE. 119 § 37. In the above paragraph I have incidentally stated the important fact brought out by Experiment B, that four pins in a square are judged better than three in an equilateral triangle of the same base. Why this should be so under our hypothesis, Avhile "common sense" would expect the contrary, I have space here to demon- strate only partially. We may anticipate that we have to do here with matters of experience conditioned by geometric arrangements. By Law Two, pairs of points are increasingly disassociated proportionally to their distance apart. The diagonal points of the square are further apart than any pair of points in the triangle. By the mere law of distance-average, therefore, the square should rank above the triangle. Ko doubt the diiference between the lengths of the hypotenuse and the sides of the square give a favorable " cue "' in reasoning out that the figure pressed on the skin must be a square and therefore has four pins. But we may note as to this, that while the difference of length between the inter-distances in that category of our experiment where four pins are arranged in a triangle — one being in the center — is greater than in the four- I)in square, yet the judgments for such an arrangement of four pins are inferior to those of the square of the same base. (Block 216, IV pins in triangle 47.20, in square 54.60.) We must consider, therefore, that the superiority of the square over the triangle in numerieaJ^:^," ih A *>; 120 OUK NOTIONS OF NUMBER AND SPACE. judgment, is fundamentally rooted in tlie integrity of Law Two carried over into the more complicated judg- ments of two-dimensioned space. In this light the whole matter becomes confirmatory of our genetic hypothesis, and highly instructive as to the intimate formation of those mental processes which are a degree more complex than the most simple ones. § 38. As bearing on the above I have only room to note in Table 15, Avhere no rocking was permitted, that the IV-pin square still shows superior to the Ill- pin triangle, while in Table 16, where the mind was forced to neglect the geometric impressions and to attend alone to the points actually felt, the averages show the III pins to be judged correctly 9.2 times in the tri- angle, and the IV pins only 8.6 times in the square. That is, when the geometric influence is shut out, the judgments fall back to the common principle that three impressions may be better distinguished than four, § 39. Turning from Law Two to Law Three we discover upon slight study that, within categories as nearly similar as could be chosen for comparison of the different arranging of the pins, the same general drift of over-estimation is manifested throughout Experiment B as we discovered throughout Experiment A ; also, over-estimation is greatest now in the same relative places as formerly, namely, in the shorter distances. These above facts are, perhaps, evidence for laws OUR NOTIONS OF NUMBER AND SPACE. 121 already redundantly confirmed, but the subject gains extended interest when we examine, from another point of view, the amount and the distribution of the errors in the B experiments. We could have expected from Laws One and Three, that the greater the number of numerical categories used in any experiment, the greater would be both the amount and the drift of the errors made under these two laws. For instance, if we made new investigations like those of Experiment A, but with numbers of pins ranging from III to IX (as in Experiment B), in place of from II to Y, as formerly, we should expect the amount and the range of errors to be much increased. Upon the face of it, the possibility of error where the judgments may range from III to IX is greater than when they are limited between II and V ; both the mathematical chances are greater under Law One, and the psychological chances are greater under Law Three. When, however, we study the results of Experiment B, we find quite the reverse of what, from the above, was to have been expected. By way of examining into this Ave must first grasp more clearly the relative amounts of error made in tlie two experiments. We had trouble in getting an exact basis for this comparison, but in rough ways we may yet get truer ideas of it. All the conditions, save those whose results we are seeking to measure, favor 122 OUR NOTIONS OF NUMBER AND SPACE. the Il-pin judgments ; that is, other things equal, II pins ought to be judged better than III. If, now, the new triangle judgments happen to exhibit a less amount of error than did the old Il-pin judgments, we may then justly take the amount of this improvement to be a partial measure of the new conditions of the experiment. For a full comparison one must go to the full tables, but we will bring forward a few test items. In the following table we will compare both the maximum amounts of error made for II pins in Experi- ment A with those for II pins in Experiment B, and also the corresponding average errors: Similarly, the total average in the general summaries (Tables 10 and 17) are for II pins + 41.6, and for the Ill-pin triangle +23.2. For a still rougher comparison, the grand averages of the same summaries give us the 1 See upper left-hand corner of Block 70, Table 4, and cor- respondingly for other tables. 2 See footing of left-hand column, Block 70, Table 4, and cor- respondingly for other tables. OUR NOTIONS OF NUMBER AND SPACE. 123 following errors: Experiment A, + 5.8 ; Experiment B, +2.1 (lower left-hand corner, Blocks 121 and 217). The superiority of the triangle arrangement is obvious everywhere without comment. § 40. The above figures having furnished us a clearer demonstration of the amount of superiority of two-dimensioned judgments over the lineal, we must look at certain differences in the distribution of these errors under the two experiments. We get at these quickest by an illustration. If we turn back to our old typical Block 65 and read the top line, we get the cor- rect judgments at 1-cm. distance for II, III, IV and V pins, respectively, as follows: 7, 32, 59, 55. We remember the explanation of this remarkable increase of apparent accuracy from left to right; that it Avas due to the drift of the great uncertainty of the short distance-judgments. If now we turn to the top line of the corresponding Block 196 of Table 14, we get, with a similar ascending series of pins, a descending series of correct judgments, as follows: For triangles 46, 32, 31; and for squares 44, 19, 28. We have no longer tlie remarkable increase from left to right due to drift of error under Laws One and Three. What is the trouble ? Are these laws inoperative here ? Xo ! but we have, in a particular but legitimate exception, a remarkable proof of the general integrity of these laws everywhere, both throughout Experiment A and tlirough- 124 OUR NOTIOISS OF NUMBER AND SPACE. out Experiment B. We have recalled that the drift from left to right in Block 65 was a drift of uncertainty). If now we look at the top line of Block 199 we see thajb the uncertainty there is very small as compared with the corresponding line of Block 70. The maximum error is for the former +32.0; for tlie latter +109.9. The average error for the former is — .3 ; for the latter + 35.7. The drift of error from left to right in the new experiment ought, therefore, to be proportional to this marked decrease in error. And so it is. Appar- ently there is none whatever, and the number of correct judgments actually decrease from left to right. It is possible that corrections for Law One would still leave a small drift under Law Three, but in any case it would be so small as to constitute a peculiar proof of the integrity of this law carried up into the more compli- cated judgments of two-dimensioned space. Of course this is but a sample of proof which would be augmented shovild we extend our examinations. For the forehead and for the forearm the amount of error is less in Experiment B than in A, but not so much less propor- tionally, as in the above sample taken from the abdomen ; accordingly, the drift under Law Three should be less than formerly, but not so much so as in our above illustration. This is just what occurred, as the follow- ing figures taken from the tables will demonstrate. They give the top lines respectively of Blocks 25, 30, 45, 50, 148, 153, 173 and 178 : OUR NOTIONS OF NUMBER AND SPACE. 125 Experiment A. Forehead. 5 + 84.2 16 61 + 43.5 + 1.5 Forearm. 64 - 8.7 26 + 70.0 31 45 + 18.1 - 11.7 Experiment B. Forehead. 23 - 27.3 26 + 69.1 28 + 35.0 54 27 + 12.9 + 53.5 Forearm. 54 + 28.9 78 - 8.9 28 + 58.2 28 + 21.4 34 34 - 9.5 + 32.4 35 + 10.3 44 - 21.5 Here, for the forehead in Experiment A, between II and V pins, we see a drift of correct judgments from 5 to 64, and of amount of error from +84.2 to — 8.7 ; while in Experiment B the corresponding drift between III and IX pins is only 26 to 78, and +69.1 to —8.9; and so similarly for the forearm. § 41. Had we space, we ought to consider further the distributing influences of our laws under the different conditions of our experiments and for the different regions of the body; but this we must now leave to the individual student. Compelled now to pass on to other matters, we may summarize our imperfect study of the number-judgments in our new Experiments B, as fol- 126 OUE NOTIONS OF NUMBER AND SPACE. lows: We observe in the new workings of our laws botli certain modifications of old traits running parallel to definite changes in the conditions under which they act, and also entirely new traits due to new conditions. We find all these manifestations agreeing with each other and conforming to the reasonings of our general hypothesis; we must, therefore, admit them to be strong evidence for its truth. DISTANCE-JUDGMENTS BASED ON TWO DIMENSIONS. (^Experiments B, C and D.) § 42. The elementary law of Association is that the resultant state at any moment is the indissoluble product of the sum of all the tendencies active at that moment. It is fundamental to all that I have hcBetofore said, that this law holds good for tlie stimulation of each and every possible combination of nerve-ends. When we stimulate a single nerve we get a specific effect which expresses the tendencies developed for that definite stimulation. AVhen two given nerves are stimulated we get a different effect, also specific, and expressing the tendencies developed coordinately for the given combination as stimulated. Just what rela- tionship the specific state (which expresses the sum of OUR NOTIONS OP NUMBER AND SPACE. 127 combined stimulation) bears to the several specific states (which, respectively, on occasions express the stimulation of each nerve separately), our science does not now with confidence suggest. We cannot yet for- mulate the coordinate tendency, in terms of tlie several separate tendencies. But while we may not determine its particulars, we may demonstrate that there is such a relationship ; and this is to be my present thesis. Stated more explicitly it is that : The actual effect of any combination of nerves is partially the resultant of the combined experiences of the given combination, and partially, also, is to be traced back to the experiences which have influenced and developed each element or possible sub-combination of elements in the total combination separately. It is the latter part of this statement which, in approaching more complicated distance-judgments, we must especially consider.^ § 43. We may profitably bring this matter before us by considering one of the triangles of Experiment C. The sides of these triangles were formed by straight edges of cardboard, the card being folded as when forming the sides of a paper box. We are to study the coordinated result of pressing one of these lineal figures iipon tlie skin. C-all the triangle ABC. ISTow, by our 1 It must be borne in niiud that we never suggest that the resultant mental state is other than an indecomposable specific wliole. 128 OUR NOTIONS OF NUMBER AND SPACE. thesis, upon the simultaneous stimuLation of the lines A, B, C, there will tend to rise every effect common to the separate stimulation of every s\ib-combination of nerve-ends possible under the laws of permutation and combination for the total number of nerve-ends in the wliole lineal triangle. We do not say that the actual effect will be the resultant solely of the sum of these tendencies. On the contrary, every time the whole triangle is affected, as in our present case, there will be left thereby a direct modification of the habit of reaction of the total combination, and this modification will, in some degree, manifest itself in all subsequent activities of the total combination. But we are now to consider the tendencies of the many possible sub-combinations. If we examine the case of any two points, y and x, in the perimeter of the triangle (one, at least, of the same not being a corner) we see, by our general, thesis, that the distance-judgments based upon the separate stimulations of any such points would be shorter than that of the sides of the triangle. Consequently, by our present thesis, the influence of all such tendencies, in the sum of tendencies resultant from stimulating the whole triangle coordinately, ought to shorten any dis- tance-judgment simultaneously based thereupon. If this resultant judgment is to estimate the length of the sides of the triangle, or of one particular side, then, OUR NOTIONS OF NUMBER AND SPACE. 129 although the "developed tendency"' corresponding to the side may be the chief factor in the formation of the judgment, yet these other tendencies, being in activity by reason of actual peripheral stimulation, they cannot be wholly got rid of and will enter into the sum total of present influence ; and, being shorter than the tendency corresponding to the full side, they will shorten the resultant judgment from what it would be, were the same distance measured simply between two pins. If, thus, our thesis is correct, all the distance-judgments for triangles in Experiment C ought to be shorter than judgments of corresponding distances in Experiment A. § 44. Before we examine our tables to test this theoretical conclusion, a relative matter must be con- sidered. It concerns the ''sharpness" of the different modes of stimulation. Up to the present, to avoid con- fusion, I have excluded the factor of ''pure sensibility" from our studies. In proper place, I shall bring in special investigation to demonstrate what I shall state here dogmatically, namely, that, within certain limits, ^'sharpness of stimulation^^ shortens the resultant dis- tance-judgment. Xow, according to this, and since the pin-points of Experiment A furnish a much sharper mode of stimulation than the card-edges of Experiment C, in comparing the judgments of the two we must bear in mind that those in Experiment A are shorter than, for a perfectly just comparison, they should be. If in 130 OUR NOTIONS OF NUMBER AND SPACE. tlie actual results the theoretical demands are some- times apparently unfulfilled, due allowance must be made ; and if they are fulfilled without such allowance, the proof of our thesis must be considered as the more marked. § 45. Toward making the proper .comparisons, I first present the following table. In the vertical columns are given alternately, the maximum and the average over- estimation taken correspondingly from the '■'■average'''' distance-judgments of Experiment A, and the triangle distance-judgments of Experiment C. I have used the "average" here in place of the Il-pin judgments referred to in the theoretical discussion, as it may be claimed that the latter are unduly lengthened by the distributing influence of Law One. The "averages," since they are free from such criticism, are in theory equally eligible to the proper comparison and, in fact, are shorter, as, by Law Four, they should be, and will, therefore, if they stand the test, demonstrate our point in hand even more profitably than would the "Il-pin" judgments: OUR NOTIONS OF NUMBER AND SPACE. 131 H + I + + O CO O lO ++ + + o c; cc -^ + + c o ^ CO + + + + + + lO o . : - 2^ CO o oc ^ ,« ^ r« o o o o o o o o pq W M K ^ ^ ^ ^ « ^ ^ J2 .a c« eS eS ce H H H H ^^ 132 OUR NOTIONS OF NUMBER AND SPACE. These figures, with the exception of those for the abdomen, confirm our theory, even without allowance being made for sharpness. And as the abdomen, owing to the thinness and tenderness of the skin, is just the region where sharpness of the impressions from the pins would be of greatest effect, as compared with the rather dull feeling from the paper triangles, I think the evidence of the table is beyond question. Particularly we call attention to the figures for the last line of the table ; these are for the '' (a) method," where the apparatus is applied evenly, without rocking. There is reason to believe that the effects under present discussion would come out purest in this method. It is, therefore, of interest to note that the contrast of the two sorts of judgments is more marked in these results than anywhere else in the table. § 46. The straight-edge judgments of Experiment A (see Tables 8 and 9, discussed on page 109) were^ shorter than those for the same distances measured by pins in straight line. Theoretically, for the reasons already given, our triangle-judgments should be shorter than either. Comparing the average errors, respectively, from Tables 8 and 9, with the corresponding figures for triangles in Tables 19 and 21, we get as follows : OUR NOTIONS OF NUMBER AND SPACE. 133 Experiment A. Experiment C. Straight-Edge. j Triaxglk. i Regular Method . . + G.9 + 5.3 Method (a) + 8.9 + 2.4 Again, without allowance for sharpness, the average errors all show the superiority of the triangle-judg- ments, and again the contrast is most pronounced in the purer (