Abstract
Vittorio Benussi (1878–1927) is known for numerous studies on optical illusions, visual and haptic perception, spatial and time perception. In Padova, he had a brilliant student who carefully worked on the topic of how people estimate numerosity, Silvia De Marchi (1897–1936). Her writings have never been translated into English before. Here we comment on her work and life, characterized also by the challenges faced by women in academia. The studies on perception of numerosity from her thesis were published as an article in 1929. We provide a translation from Italian, a redrawing of its 23 illustrations and of the graphs. It shows an original experimental approach and an anticipation of what later became known as magnitude estimation.
Keywords
Vittorio Benussi was born in Trieste in 1878, when this city was part of the Austro-Hungarian Empire. He studied at the University of Graz, where he became assistant of Alexius Meinong (1853–1920), then untenured professor from 1906 and, until 1918, head of the Hapsburg Empire's first laboratory of psychology, founded by Meinong in 1894. When in 1919 he obtained the chair of Experimental psychology at the University of Padua he brought innovative research methods to the Kingdom of Italy, as well as Gestalt school ideas and debates. He had many interests, from basic research on perception of shape and time, to the study of suggestion and hypnosis and applied psychology (Antonelli, 2018; Bobbio & Giora, 2019; Giora & Bobbio, 2022).
Silvia De Marchi was studying Literature when, in 1920, she started to attend Benussi's lectures and then the activities in his laboratory (Bobbio & Giora, 2023). In 1924, she obtained a degree in Philosophy defending an experimental thesis (De Marchi, 1923–24). Her empirical work was the only one Benussi supervised during the nine years he spent in Padua (Bobbio & Giora, 2023; Bobbio & Roncato, 2023).
In 1925, De Marchi was appointed voluntary assistant, an unpaid position at the very first level of the academic career, the second assistant that Benussi trained after Cesare Musatti (1897–1989). Notably, in a university environment dominated by men, Benussi made room for a woman. At the time (1925) no woman had a formal faculty position at the University of Padua. Only about 7% of those with a teaching or research assistant role were women and worked mainly in the Faculties of Medicine or Science. De Marchi was the only woman assistant in the Faculty of Philosophy and Humanities. After Benussi's tragic death (by suicide, in 1927), she continued cooperating with Musatti, whom she married in 1932. She died after complications of ear infection in 1936. Unfortunately, since childhood, her life was weakened by health problems until this premature ending (Bobbio & Giora, 2021, 2023).
Even in these early days of the history of Psychology, some authors had written about perception of numerosity. In particular, within the Gestalt tradition, there was a view that quantity could be perceived as an emerging property (Liebenberg, 1914; Nanu, 1904; Pegrassi, 1904; Ponzo, 1928). Mario Ponzo (1882–1960) was an Italian psychologist, a pupil of Federico Kiesow (1858–1940) who worked first at the University of Turin and from 1931 at the University of Rome. He is best known for a configuration he examined and that now bears his name: the Ponzo illusion. In a paper published in German in 1928, Ponzo used the effect of misperception of size as a tool to explore perception of numerosity. The effect now known as the Ponzo illusion was presented as a case of size contrast, along with other cases where when people misperceive extent, they misperceive also numerosity (Ponzo, 1928).
Recently, a paper by Ponzo (1928), originally published in German, has been translated into English (Bertamini & Wade, 2023). This paper appeared also in Italian in 1929, in Issue 7 of the Archivio Italiano di Psicologia, pages 1–37. This is the same issue in which we find the main paper on numerosity by Silvia De Marchi. The page numbers for her article are 177–225. As a personal note, when the translation of Ponzo's work was published, some colleagues contacted one of us (MB) pointing out that the paper by Silvia De Marchi was equally important and should not be forgotten. This was in part the motivation for the present translation.
After a robust theoretical premise, the thesis by De Marchi (1923–24) had three empirical sections. The first one dealt with the diagnosis of facts (Tatbestandsdiagnostik) and followed Wertheimer and Klein's (1904) approach in which free associations were used as a tool to distinguish truth-telling from lying. The second dealt with the respiratory symptoms accompanying the experience of telling lies, detected with Benussi's method for computing the quotient of inhalation to exhalation using the pneumograph (Benussi, 1914). The third section investigated perception of collectivity, an important experimental issue in itself. In the context of forensic psychology, it was relevant to understand inaccurate reports occurring even with simple judgments, like the numerosity of configurations of dots (Bobbio & Giora, 2023; Bobbio & Roncato, 2023). We prefer to use here the term “numerosity” instead of collectivity although in the original writings by De Marchi the Italian word “numerosità” was not present, probably because other terms are more general and less directly linked to mathematics and numbers (collectivity, quantity). A large literature on perception of numerosity has developed in more recent years. De Marchi states clearly that she is studying how the numerosity of simple elements, like dots, can be estimated (“valutazione”) at a single glance, without counting. The assumption is that observers have direct impressions of numerosity, an idea closely related to the existence of an approximate number system. This mechanism is believed to allow estimation of sets of elements (Anobile et al., 2014; Burr & Ross, 2008; Dehaene, 2011), and to provide the building block for numerical development in general (Cantlon & Brannon, 2006; Dehaene, 2009; Piazza, 2010).
Benussi had already started exploring this topic while working at the University of Graz, approximately from 1908 (Antonelli, 2018; Bobbio & Giora, 2019; Bobbio & Roncato, 2023). In Padua, he handed over the investigation on the perception of numerosity to De Marchi around 1922, similar to what he had done with the stereokinetic effect, which was described by Benussi around 1918 and later assigned to Musatti (Bobbio & Giora, 2023; Giora & Bobbio, 2022; Musatti, 1924, 1975). A recent review of stereokinetic phenomena is in Zanforlin (2017).
De Marchi showed a genuine talent for experimental research, in a way that was particularly aligned with Benussi's style (Bobbio & Giora, 2023; Bobbio & Roncato, 2023). Although she worked on various projects, her contribution to the study of numerosity stands out for its originality (Bozzi, 1969; Luccio, 1983, 2017; Marhaba, 1981). Indeed, the procedure developed by Benussi and De Marchi anticipated the method of magnitude estimation (Agostini & Luccio, 1994; Masin, 2006).
De Marchi starts from the observation—which fits Benussi's differential psychology approach—that a tendency to overestimate or underestimate is a stable characteristic of observers, that is, people tend to be consistent. Moreover, she notes that the estimation is largely unaffected by the knowledge of the exact number, because observers have a direct impression of numerosity, and this perception is different from counting or reasoning about numerosity.
She refers to experiments as “esperienze” (experiences), the participants are involved in collective observations, the materials and methods are not described with the details that are expected in modern literature, and the instrumentation is ad-hoc. However, De Marchi was able to identify several key factors that affect estimation. In particular, she manipulated extent (the surface over which the dots are located), configuration, density, and exposure. With respect to time, she noted that large numerosities tend to produce a subjective temporal shrinking. These factors are broadly consistent with what has emerged in the more recent literature. With respect to time, De Marchi speculates that subjective time decreases with the difficulty of the estimation, and this effect of task difficulty has been confirmed in many studies (e.g., Buhusi & Meck, 2009; Zakay et al., 1983). There is also evidence that perceived numerosity increases with element size (Ginsburg & Nicholls, 1988; Shuman & Spelke, 2006), regularity (Bertamini et al., 2023; Ginsburg, 1976), and for larger areas (Dakin et al., 2011; Krueger, 1972; Tokita & Ishiguchi, 2010; Vos et al., 1988).
The results about spatial arrangement and grouping are complex. De Marchi is aware of this complexity and the fact that different factors interact. For example, for small areas, as density decreases estimated numerosity increases. For large areas, a more sparse/less dense stimulus leads to a decrease in estimation. With respect to grouping, she notes that when observers perceive the configuration as having subgroups, they will tend to underestimate. Recent work has found that creating subgroups, for example with different colours, leads to underestimation (Chakravarthi et al., 2023; Poom et al., 2019). Similarly, clustering of elements, by proximity, leads to understimation (Bertamini et al., 2016; Valsecchi et al., 2013).
As a further example, De Marchi (1931) reported a study based on the Müller-Lyer illusion, raising the possibility that different results may come from the qualitative versus quantitative approach to the perception of numerosity, contrasting Ponzo's approach to hers. Ponzo asked observers to compare two pictures with the same number of dots, with the assignment of choosing the one where the dots appeared more numerous. Instead, De Marchi asked participants to estimate the number of dots in single configurations, shown for a limited time so that counting was made impossible (tachistoscope method). The comparison task (dichotomous) or the evaluation task (quantitative) elicited different perceptual and cognitive processes, the former being more dependent on the differences between the two configurations (De Marchi, 1931). This dissociation has been confirmed experimentally by Alam et al., (1986), and by Agostini and Luccio (1994). For the theoretical discussion see also Bertamini (2023) and Luccio (2017).
Early work on numerosity has largely been forgotten, mainly because of a language barrier, and we hope it can be more fully appreciated, including the contribution of researchers like Silvia De Marchi one hundred years ago. In her short-lived career she produced careful and original works that foresaw some themes that are still being investigated today.
Translation
This journal article is based on the thesis completed in 1924 at the University of Padova. Preliminary results constituted an oral presentation at the 6th Italian Congress of Psychology held in 1923. The thesis was titled “Contributi alla psicologia giudiziaria” [Contributions to forensic psychology] (De Marchi, 1923–24). The reference of the journal article is: De Marchi (1929). Le valutazioni numeriche di collettività. Archivio Italiano di Psicologia, 7, 177–225. The original numbering of paragraphs and subparagraphs has been omitted. All terms in Italics in the original paper are preserved. Figures and graphs have been redrawn for clarity and they retain the features of the original article. Figure captions have been added and were absent in the original. Footnotes have been renumbered, the references formatted in APA style, and are listed at the end of the text.
(Psychology Laboratory of the Royal University of Padua founded by VITTORIO BENUSSI).
Introduction
Preliminary Findings
If we show different subjects a group of dots as in Figure 1 (dots = 158) for a very short time, so that the possibility that the subjects count even a part of it is excluded, and we then invite them to evaluate with a digit the numerosity(*) seen, we can see that the judgments obtained are subjectively certain (the response of a subject fluctuates for example from 70 to 80, from 120 to 140, from 200 to 320, but within these limits they are confident), but they vary considerably from subject to subject.
A configuration with 158 dots.
It can be seen that the evaluations obtained in some cases are higher than the real number of dots, in other cases they are lower. 1 In short, we obtain overestimations or underestimations which are true “individual constants”. 2 Whoever, for example, underestimated a stimulus once, underestimates again when faced with new stimuli. Thus, underestimations and overestimations correspond to typical qualities of the subjects, they are not accidental phenomena but, as we said, individual constants. 3
The spatial arrangement, the grouping, and the form 4 in which equal collections are perceived, are of fundamental importance in the evaluation. Often these factors seem to be more powerful than the individual characteristics. However, the type of under- or overestimator remains stable, if not in absolute data, then in relative data. So, for example, a specific formal factor can bring the evaluation of an underestimator from 20 to 25, that of an overestimator from 100 to 125, while the number of elements is equal to 50 in both cases. Overall, therefore, there is a (relative) overestimation due to the influence of the formal factor.
The importance of these formal elements can be seen by presenting figures like those in Figures 2–6 for a very short time, 5 so as to exclude any possibility of counting.
Three configurations with 55 dots arranged within different surfaces.
Two configurations with 98 dots arranged within different surfaces.
Two configurations with 98 dots arranged with different forms.
Two configurations with 69 dots arranged with different forms.
Two configurations with 90 dots arranged with different forms.
I will report some qualitative and numerical data.
If an equal number of elements are first arranged on a circular surface, and then on a rectangular surface, we observe that:
up to a given ratio between the sides of the rectangle there is (relative) overestimation compared to the elements of the circular surface; beyond this given ratio there is an underestimation of the elements of the rectangular surface compared to those of the circular surface (see Figure 2).
The dots of the three groups in Figure 2 (objective number = 55) are evaluated, by the same subject, as 40 when they are placed on a circular surface, as 50–54 if placed on a rectangular surface. Instead, the two groups of Figure 3 (objective number = 98) are evaluated by the same subject as 58 when they are arranged on a circular surface, and as 54 on a rectangular surface.
The two groups, amorphous and stellar, in Figure 4, are evaluated respectively as 74 and 83, while the objective number is the same as that of the previous figures (98), which had been evaluated as 58 and 54 dots. Moreover, the two groups in Figure 5, made up of 69 elements, are evaluated as 42 if arranged as the group on the left, and as 80 if arranged as the group on the right.
Of the two groups in Figure 6, both consisting of 90 dots, the one on the left is evaluated as 50, the one on the right as 98. This response was from the underestimator subject who said 58 for the circular configuration with 98 elements in Figure 3.
From these simple observations, it is clear that form is important. Moreover, there is variation in the responses corresponding to variation in form.
General Theme: Evaluation Factors
The few notes mentioned above are sufficient to indicate the fundamental questions that the experimental study of numerosity must try to resolve. And precisely:
what determines the type, or rather by the action of which factors one subject overestimates and another underestimates (considering the evaluation both in a general and absolute sense). what and how many factors determine a relative under or overestimation regardless of the type.
The present study therefore aims to analyse the evaluation factors.
6
Evaluative situations result from multiple determinants, some of which are external, that is, given by the objective conditions of the stimuli, while others are internal. These constitute the subject's conscious behaviour in front of the stimuli. Among the external determinants we can consider:
the duration of exposure; the size of the surface occupied by the elements (i.e., dots); the density of the elements presented; their spatial arrangement; their arrangement in time: depending on whether they are successive or synchronous; the arrangement in space and time, depending on whether the elements are at rest or in motion; the arrangement in space and time, depending on whether their succession is fast or slow; and depending on whether their movement is fast or slow; attentional behaviour: (a) attentional setting; (b) concentration or attentional disposition; mental fragmentation; reflective processing; absolute impressions of multiplicity (abundance?) or poverty (scarcity?); immediate or mediated formal connections; persistence of consecutive mindfulness.
Among the internal determinants:
Analysing the evaluation process means, in short, finding the role that these factors have on the evaluation. This process must be distinguished from that which can lead, from the knowledge of one's own errors, to learning to evaluate precisely. The analysis of this second process would constitute not “the psychology of evaluations” but “the psychology of learning a particular behaviour appropriate to reality.”
7
Here we consider the laws of evaluative errors in the case in which one has not yet learned to evaluate adequately, and it is not possible for the subject to learn because they are never made aware of their errors. Having thus posed the general theme, I move on to the technical aids.
The Devices
There are two devices used: one for evaluations of configurations at rest, for static experiences, and the other for evaluations of configurations in movement, for dynamic experiences.
The Ernemann projection device was used. Lens, which can be moved within very vast limits, allows projections of several meters and a few centimetres on a side. The size of the surfaces of the stimuli is varied by varying the position of the device, that is, the distance between the slide and the transparent screen on which the stimuli were are projected.
A kymograph(**), which rotates at a uniform speed, carries an insulating mantle in which slits of varying widths are cut; a metal pen is in contact with the mantle itself so that through one or the other of the slits it can touch the metal part of the kymograph for different times, corresponding to the width of the slits.
The lamp of the projector is operated by a circuit which passes through the metal pen and the kymograph, so that illuminations of the slide, and therefore projections of the figure on the screen, of different durations are obtained. The duration itself can be varied by varying the rotation speed of the kymograph cylinder. Exposures of 200, 400, 800 ms and more can thus be obtained.
In the dimly lit environment where the research or experience is conducted, the kymograph is set in motion and, a few seconds before the exposure of the figure, an agreed signal is given: (attention!) and immediately before the exposure a second signal is given which indicates the imminent appearance of dots: (now!).
The subjects have the task of looking carefully at the screen, embracing it with their attention in a uniform way. The procedure of the experience will be explained further below.
The second device concerns the experiences of stimuli in movement. The slide is fixed, in this case, on a mobile frame that slides across a counterframe. The latter is covered at the front by an opaque plate bearing a slit; it thus projects a luminous strip onto the screen on which the dots of the slide appear when the mobile frame moves across the counterframe. The observer therefore always has new dots in front of him which are seen moving through a more or less wide exposure field depending on the width of the slit.
The frame that carries the slide, moving, first opens and then interrupts a circuit corresponding to the moment in which the dots begin and end. The circuit activates an electromagnetic pen that marks the beginning and the end of the dot movement on the smoked coat of a kymograph. A Jaquet chronoscope marks fifths of a second above the line traced by the electromagnetic pen. Thus, the overall duration of the exposure of the moving dots can be measured without difficulty. The speed of the moveable frame, and therefore of the dots the subjects see, is regulated by a second kymograph.
Speeds corresponding to movement durations equal to d1, d2, d3 were used, in which:
Now considering that in the large field projections (see paragraph The area) the slit corresponded to a field 5 cm wide, the following durations are obtained for the individual dots, relating to the unit of space of one centimetre: d1 = 65 ms, d2 = 115 ms, d3 = 187 ms.
The corresponding speeds are:
15.4 cm per second for d1, 8.6 cm per second for d2, 5.3 cm per second for d3,
Having briefly exposed the technical aids used in our experiences, we move on to consider the general points of the method followed.
The Method
The groups of dots were presented with the tachistoscope method. The exposures must in fact be very short to prevent the subject from reaching the solution by “counting” the elements of all or part of the group. If they could do this, there would no longer be a numerical evaluation: that is, that process by which an aggregate perceived in conditions that cancel out any possibility of numbering its elements is expressed in figures.
With respect to the behaviour of the subjects, the experiences were divided into purely evaluative and introspective. 8
In the first case, the subject had to record in the protocol the first, unreflective evaluation that arose in his mind as soon as he saw the projection, without any subsequent evaluations, based on impressions elaborated afterwards, nor on particular subjective aspects of the behaviour preceding the writing of the protocol.
In the second case, however, it was the subject's task to record a detailed introspective protocol on the immediate evaluation, on the phases that had preceded, on the mental elaborations that followed the first evaluation and on any new evaluations that arose following reflection, in comparison with previous experiences, etc.
This second group of experiments was carried out after the first so that the subjects, made familiar with the experiences, were ready to analyse their own internal behaviour and to specify introspective internal data that less experienced subjects might not have done. These two methods of: immediate pure evaluation and reactive introspection allow us to process the data obtained separately; first the numerical evaluative data regarding the different external and internal conditions of the immediate pure evaluation experience: then the introspective data that serve us to integrate the numerical-evaluative results into their different conditions of experience.
In the present work, we mainly consider the results of the immediate numerical evaluations, limiting ourselves to those references to the introspective data that are necessary for an interpretation of the quantitative data. A more exhaustive discussion of the introspective data will form the subject of a second study.
Before these two groups of experiments, two series of experiments on “absolute collective impressions” were carried out.
Previous Research
Before exposing the results obtained from our experiences, I briefly consider a few data from previous research. These data are mainly due to Liebenberg 9 and Benussi. 10
Liebenberg (and more recently Mokre) 11 researched the laws of valuation for very small groups. Liebenberg’s material does not exceed 18 units, these are arranged on horizontal or vertical straight lines or on curves. They are of various colours and are grouped three by three or four by four. He notes that groups from 5 to 7 are exactly evaluated if they are exposed tachistoscopically, the others are overestimated the more the more the number of elements increases. Considering these observations, we can conclude: (a) that the situation studied by Liebenberg is not a true evaluation situation, but a situation similar to that of “counting,” (b) that a pure evaluation is not obtained with his system because it is based on exercise. We, however, as we said, are not interested in the learning of a relationship between the impression of numerosity and the number, but the spontaneous correlation based, on the one hand on the pure impression of quantity, and on the other on numerical impressions.
It is evident that, when the elements are few, it is possible to count them, if not in the fleeting moment in which they are exposed, then during that impression that follows any perception. The numerosity remains, even after some time after its disappearance, “mentally present” 12 and in front of this mental presence we can behave as if in front of a perceptually present object.
Experience teaches us that, when the elements are aligned on a straight line or a curve, they tend to be subjectively grouped into clusters of three or four. The number of objectively given elements is probably overestimated because partial enumerative processes are included in the evaluation. Generally speaking, Liebenberg did not analyse the importance of form and movement factors.
The value of the structural elements was instead understood by Benussi (from whom the examples shown in (a) were taken). Indeed, the present research is linked to his experiences and seeks to analytically develop some points of view that had been put forward.
The Absolute Impressions (of pp, p, i, m, mm)
The object of the evaluations that we must now analyse is a “magnitude.” As such it is subject to the influence of impressions of absolute magnitude and to the law of relativity of impressions of magnitude in general. Let us now try to clarify the meaning of the words “impression of relativity” and “absolute impression”: we are in the first case when we find ourselves faced with objects which, compared to each other, give an impression of “bigger,” “smaller,” “the same.” However, the same objects can sometimes also give impressions of “large,” “small” not based on references to an ideal unit of measurement nor to any term of comparison. In this case, we are faced with an absolute impression of greatness; and all objects that have size can arouse in us absolute impressions of size, independent of any attitude of comparison. 13
We precede the experiences of numerical evaluations with some data on absolute impressions. The perception of numerosity can animate absolute impressions of a lot, or of a little, and it is also possible that the numerical evaluation is based on this absolute impression. It is possible for example that under- and overestimator types base their evaluation on absolute but opposite impressions of numerosity.
These absolute impressions are dependent on the general approach of the subject.
For example, if the setting of the subject is given by the representation: “newspaper,” and if the subject is presented with a newspaper of 80 cm in height, that newspaper will give the subject an absolute impression of “large.” If instead of a newspaper, the surface of an 80 cm long “table” is presented, the subject will have an absolute impression of “small.”
In our experience, given a surface and a particular density of dots or disks contained within it, it is evident that, depending on the relationship between the size of the dots and the size of the surface, the impression of “many, few, very few etc.” will vary even if the number of elements stays the same.
We must therefore establish: (a) the objective number of dots that causes, for a given surface, an immediate and pure impression from any other situation, of “many,” “very many,” “few,” “very few,” “unspecified”; (b) the number of dots that a subject mentally deems necessary to have an impression of very many, many, unspecified, few, very few: (mm, m, i, p, pp).
Part A. In the first experiment, the subjects (who were 8) were shown, for a duration of 280 ms, groups of dots arranged with different densities in the different exposures, on a constant surface. They were given the task of evaluating the numerosity with the following expressions: the dots are very many (mm), or the dots are many (m), or unspecified (i), or few (p), or very few (pp).
The 23 dot groups, ranging from 5 to 162, were exposed to the subjects 8 times. Based on the results obtained, for each subject, Table 1 was created, in which the percentage of evaluations (mm, m, i, p, pp) is marked, relative to the objective number of dots.
On the basis of these data, therefore relating to all the subjects, Table 2 was constructed in which we label the evaluation (mm, m, i, etc.), whose average percentage among all the subjects is the highest.
From Table 2 it appears that, by integrating and rectifying the data obtained with the usual methods, with negligible oscillations, a group of 7 elements or less is judged to be composed of very few elements (pp): that a group of 10 elements is intermediate between the very few and the few (p-pp); that a group of 15 elements is considered to be made up few elements (p); that a group of 20.5 elements is considered as intermediate between the few and the unspecified (p-i); that a group given by 27 elements is unspecified (i); a group of 45 dots is intermediate between the unspecified and the many (i-m); while one made up of 57 dots is considered to be composed of many (m) elements; that one made up of 67 elements appears intermediate between the many and the very many (m-mm); and finally the one made up of 102 or more dots appears to have very many dots (mm).
By graphically grouping these data we have the graph of Figure 7.
Trend of average percentage of responses (from very few, pp, to very many, mm) relative to the objective number of dots.
We can consider the numerical values corresponding to the responses pp (7), p (15), i (27), etc. as constituting a geometric series in which the value 1 is given to the group which, with a frequency of 100%, gives the impression of a group of very few elements (pp). This fact allows us to determine the connection between the individual absolute impression, when two of these are given (e.g.,: pp and p). The other impressions can therefore be calculated independently of the experience. If we now consider the two extremes, pp and mm, we see that we have in i a quality which, subjectively, is equally distant from pp and mm. Thus, the quality p is equidistant from pp and i, and the quality m is equidistant from mm and i.
A succession of subjectively equidistant impressions therefore corresponds, in the objective conditions that cause those impressions, to a succession of “stimuli” which is ordered according to a geometric series. This is a general law of psychophysics and our findings are nothing more, in the final analysis, than a particular case of this law.
The material of our experiences on absolute impressions is too scarce for us to make use of it in the determination of the over- and under-estimator type, but we can, for now, suppose that such absolute impressions remain unchanged when faced with the different types.
From the data obtained, however, it can be noted that two types are distinguishable among the subjects: (a) the one who prefers the “unspecified” evaluation (this can be considered as a solution of non-commitment to the task), (b) the one who, avoiding the “unspecified” evaluation prefers the others, and is therefore more certain in selecting the impression obtained.
The distinction between these two types will help us interpret the data from the experiments that follow.
Part B. We are faced, in these experiences, with a task opposite to the previous one. In the first case, the subject was invited to evaluate with the expressions: few, very few, many, etc. group of elements of which he was unaware of the number. In this second case, instead, they are invited (showing them a given surface area and the size of the elements) to say how many of these elements would be necessary to give the impression of: very many, many, few, etc.
The experiments were carried out according to the following procedure. A dark surface, limited by luminous lines, was projected onto a screen; only one dot was visible on the surface; both the surface and the dot remained constantly exposed to the subject. Then, following a random order, these questions were asked to the subject: “How many dots would there have to be on that surface to be very few?” “To be many?” “To be few?” “How many to be intermediate between the very few and the few, between the unspecified and the many, etc.?” The whole series of questions was repeated twice. Each time the subject recorded the number that seemed to them a subjectively satisfactory solution to the task. The subjects were the same 8 as in the previous experience.
While I reserve for another occasion the analysis of the reaction times necessary for the various subjects to solve the task, I move on to the data obtained, collected in Table 3.
In it I indicate with B the overall average data, with Bb the average data relating to those subjects who, in previous experiences, were of type (b), with A the data relating to previous experiences (A). The table gives us the curves of Figure 8.
Trend of average percentage of responses (from very few, pp, to very many, mm). A is the same data as in Figure 7. B is for all the responses to the new task. Bb is from the subjects who were more confident in their responses.
From these data it appears: (1) that in task B the numerical values corresponding to the expressions p, i etc. grow more rapidly than in task A; (2) that, if in task B the subjects of type (a) are neglected (that is, those who prefer evaluations of the unspecified type) and take into account only the subjects of type (b), we have data that are similar to the data of task A.
However, we can observe that even for subjects of type (b) the figures representing the solution to the task increase more rapidly than what happens in a geometric series. This probably means that the “presence of the dots” represents a factor that facilitates the impression of mm, m, i. That is, an objective number (situation A) of dots lower than the imaginary one (situation B) is necessary to have impressions of very many, many, unspecified.
We can perhaps consider the importance, now noted, of the factor presence of the dots, which can also be called “perceptive liveliness or vividness” as due to that impression of “imposing itself” that certain objects of perception can animate in some subjects. One of the most important determinants of the under- and overestimator types may perhaps be found in the predisposition to a specific perceptive impressionability.
Density in Amorphous Groups
We call “amorphous group” a configuration whose elements are arranged in such a way as not to give any particular impression of shape, and cannot be grouped under any architectural scheme. The dots are arranged homogeneously on a surface, without giving any impression of shape. The surface on which they appear is constant (30 × 40 cm with the longest side vertical); instead, the objective number of dots varies, and therefore the density of the dots themselves on the surface.
The exposure duration is, in the present series, 280 ms.
The task of the subjects (5 in this series) was to communicate in writing the number which corresponded most exactly to the impression they had from the group. The subject could also put another corresponding answer in brackets, no longer corresponding to the first unreflective evaluation, but to a subsequent elaboration. Twenty-three configurations were used within the limits of 5 and 162 dots.
The results obtained can be summarised as follows:
Subjects can be distinguished into underestimators and overestimators: a distinctive element that will be constantly repeated in all experiences and which can be said to be “constitutional” in the subjects themselves. Underestimators have a slight tendency to overestimate groups of fewer than 20 elements. The average between under- and overestimators does not come close at all to the objective number, except in its last stage. This result is of particular interest for applied psychology; in fact, it shows that an average between evaluation data of this kind, obtained with different subjects, even if the data are inadequate in opposite directions and within vast limits, does not give us reliable evidence. As the number of elements increases, both the absolute overestimation and the absolute underestimation of the over and underestimators increase.
Table 4 shows the numerical data. From this table I obtain the diagram in Figure 9 in which the objective number of dots is shown on the abscissa and the subjective evaluations on the ordinate; the α and β curves represent the average subjective data of the underestimator and overestimator type, (α + β) / 2 the average value between the two types, the straight line shows the trend of what would be the correct response. We can see that, while the α line maintains an almost uniform trend, the β line presents very strong irregularities.
Trend of average evaluations of 23 groups of dots made by underestimators (α), overestimators (β), the mean of both types [(α + β) / 2], and contrasted with those of correct answers.
This difference in performance is partly explained by the fact that the evaluative fluctuations of the underestimators occur between necessarily narrow limits (in our case between 5 and 80), while those of the overestimators occur between very broad limits (between 12 and 250).
Exposure Duration
We now expose some series of experiments carried out with the aim of determining the influence exerted on the evaluation by varying the exposure durations of the groups. Four series of experiments were performed; in them, amorphous groups were projected onto a large area (60 × 45 cm).
In the I series the exposure lasted for 280 ms (= 5 mm of the mantle covering the kymograph)
In the II series of 560 ms (= 10 mm)
In the III series of 1680 ms (= 20 mm)
In the IV series of 2800 ms (= 50 mm)
There were 23, 10, 16, and 7 subjects respectively.
In Table 5, the evaluation data corresponding to the objective number of dots is displayed. I collect these results in the diagram in Figure 10 so that they appear more easily understandable.
Trend of average evaluations of 14 groups of dots as a function of 4 exposure durations (I, II, III, IV).
From the consideration of the data, one result immediately stands out: while the exposure durations progressively increase, from the series I to IV, the evaluation data corresponding to the four series are not grouped in any direction, progressive or regressive, but are collected in two groups, so that the data of the II and IV series can be distinguished from those of the I and III: the exposure of 280 (I) and 1680 ms (III) determine a not very strong absolute overestimation, and the exposure of 560 ms (11) and 2800 ms (IV) instead determine a strong absolute underestimation.
Exposure duration constitutes, as can be seen, a new factor of relative under- and overestimation which can be added to that of perceptive liveliness or vividness noted previously.
I collect in the diagram in Figure 11 the oscillations to which it is exposed, for example, the evaluation of the numerosity composed of 162 elements when the exposure time varies from 280 to 560, to 1680, to 2800 ms. It seems that, as regards the “duration of exposure” factor, a law of oscillation occurs; that is, that there are durations that favour relative overestimation and durations that favour relative underestimation, and that these durations alternate following a specific rhythm. Further research carried out by varying the duration of exposure more should allow us to analyse the succession of this rhythm more closely; and perhaps it will also be possible to establish a relationship between this rhythm and that value of 600–700 ms which has a particular importance in perceptive processes, as it corresponds to the time required for a perceptual process to fully develop. 14
Oscillations in the average evaluation of 162 dots exposed at 280 ms (I), 560 ms (II), 1280 ms (III), and 1680 ms (IV).
Let us now briefly consider the results of one particular evaluation; that is, the evaluation of the subjective durations of exposure in the four series already considered.
The subjective duration of individual exposures. The experiments relating to the determination of subjective durations were carried out in this way:
once the series of evaluation experiences had been carried out (exposure IV, lasting 2800 ms), the subjects were given the task of evaluating relatively and retrospectively the duration of the individual exposures; then, the same white surface without dots was projected for the duration of 1680 ms, and the subject was given the task of specifying the relationship between the duration of this exposure and that of the individual exposures of series IV; finally the same white field was exposed first for a duration of 280 ms, then for a duration of 2800 ms, asking the subjects to evaluate the size relationships between the two durations.
The results of the experiments were the following:
The exposure is subjectively shorter the greater the number of dots. This finding is important because it allows us to suppose that the evaluation is the result of multiple nuclear processes.
The analogy of this result with others obtained in other fields is evident. Thus, we know that a rhythmic series seems faster if the mental work necessary to grasp its components increases. The impression of duration is therefore relative to the attentional appeal exerted by the content of that duration; the greater that is, the greater will be the “subjective contraction of time”; this result can be connected with the law relating to the relationships between apparent duration and attentional focus (15): it allows us to assume that the largest communities groups are perceived as made up of smaller subgroups spread over time.
The less successful a subject is in carrying out an initiated mental operation, the shorter the given time will seem to him. This is within the context of minimum times no longer than a few seconds. It is therefore natural that all subjects, just as they possess subjective tones or rhythms, also possess subjective “perceptive speeds.” It is thus possible to admit that, by allowing a greater number of partial perceptual processes to be completed in a given time, spread over subjective time, perceptive speed represents a factor of overestimation.
I report some evaluations removed from the introspective protocols:
C.M. subject: “the longest exposures concerned the least numerous groups”; N.P. subject: “there is something like an acceleration if the dots are more”; M.D.O. subject: “the exposures were not equal. The longest one seemed to me to be the second exposition (15 dots) in which I was able to count the dots with probability of accuracy”; S.D.M. subject: “longer time when the number of dots was lower.”(***)
2. The exposure duration of the white field (of 1680 ms) appears shorter than the exposure durations of the groups up to 47 dots (of 2800 ms); the group of 72 dots appears to last approximately the same time as that of the white field, the others are all judged shorter. The subjective contraction of time is, therefore, already in this case, very strong, since it corresponds to approximately 1120 ms. Of particular interest are the “clues” which could be more fully specified with particular investigation material concerning the two over- or underestimator types.
3. The ratio between the subjective duration of a numerosity exposed for 280 ms and one exposed for 2800 ms is on average not 1 to 10 but 1 to 20. This is another result that should be studied in more detail.
The Area
To the series (a) of evaluations obtained on a large area and for a duration of 280 ms we contrast a series (b) of evaluations of equal communities exposed for an equal duration but on a much smaller area, (7.5 × 10 cm).
I compare in Table 6 the average results in ms obtained in the two series a and b. The subjects in series a were 23, those in series b were 10: over- and underestimators were equally distributed in the two series.
We assemble these results in a diagram (Figure 12) to facilitate comparison. The result is clear: the reduction of the area leads to a subjective reduction in the number of elements; the small area corresponds to an absolute underestimation in the underestimators, and to a relative in the overestimators.
Trend of average evaluations of 14 groups of dots exposed at 280 ms on a large (a) vs very small area (b).
The crucial factors of this underestimation are probably given:
from the absolute impression of smallness relating to the area, which is transformed into an impression of scarcity (of “little”) relating to the dots; it seems that the density of the elements decreases as the surface on which they are arranged decreases. by the greater unity that the configuration with a small area has compared to that with a large area, a unity which by making the elements “cohesive” hinders that split into partial groupings which, as we have seen, is a factor of overvaluation. To check these results, a kinescopic experiment was carried out, applying, for our case, the kinescopic method used by Benussi in the field of the Panum phenomenon.
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In these experiments, two groups of dots identical in number, arrangement and clarity were projected alternately onto the usual screen, but such that one of the groups was a smaller reproduction of the other, according to the ratio between the large area and the small area of the previous experience. The alternation of the groups was such as to produce the impression of an apparent movement in which a single group became smaller and larger.
It can be observed that the impressions aroused by the alternation of the groups do not conform to those experienced when the groups are given in a static way and separated from each other. When the large area is connected to the small one in a unitary way in the kinetic transformation, the dots occupying a large area appear less “dense” than those occupying a small area.
There is therefore a relative overestimation of the elements of the smaller area: this result is contrary to that obtained in previous experiments. This kinescopic experience, therefore, cannot specify the reason for the results previously obtained, although it opens up new research possibilities in the field of evaluative situations. In this particular situation of apparent movement, there are probably factors different from those that underlie the evaluation of static groups.
Grouping and Objective Arrangement
Contour and surface. We report under this title some experiments in which: a given number of dots is arranged linearly along the periphery of a circle (contour), or in which that given number of elements is spread in a homogeneous way, as far as possible, within the surface enclosed by that circle (surface).
Let us consider in particular the cases shown in Figure 13. All groups were exposed for a duration of 280 ms, groups 4 and 5 also for a duration of 1680 ms.
Five configurations with 18 (n. 1, n. 2), 22 (n. 3), and 98 dots (n. 4, n. 5) arranged with different forms.
The subjects (4) are the same for all experiences. The average evaluation is given in Table 7.
From this it follows:
The rectilinear arrangement represents a factor of overestimation compared to the curvilinear arrangement. The arrangement on a circular surface represents, compared to the arrangement along a circular line, circumscribing that surface, a factor of underestimation. Therefore, by arranging the dots as in 1, 2, and 3, greater and greater underestimations are obtained. What applies to the linear circular arrangement and the homogeneous arrangement on a circular surface also applies to rectangular rectilinear arrangements (4) and homogeneous arrangements on corresponding rectangular surfaces (5). It is therefore important to note that an empty area does not constitute a factor of underestimation, as it might appear; that impression of emptiness therefore does not give rise to impressions of the “few” nor does it encourage underestimation. By increasing the exposure duration up to 1680 ms, the evaluations relating to 4 and 5 represented by the values (for 280 ms) of 72.5 and 45.6 rise to 122.5 and 73.5. If the first two evaluations are 1:0.64 to each other, the other two are 1:0.60 to each other. Therefore, the relative undervaluation, due to the grouping, remains constant, independent of the relative overvaluation which corresponds to the increase in duration. (b) The dimensional factor. Let us now consider the results obtained from experiments carried out in particular conditions, namely: with groups uniformly arranged on circular, rectangular (in which the ratio of the sides was 1: 1.5) and square surfaces, with approximately constant density and number of elements (and therefore variable area).
If we compare these results with those in Figure 10, we find ourselves faced with a new problem: the problem which concerns the influence that the relationship between exposure duration and spatial arrangement has on evaluation. A problem that we are content to enunciate, leaving the task of specifying its terms and results to new particular experiments.
Table 8 collects the average values obtained with 10 subjects. They were presented with groups of 72, 102, 142 dots on a circular area, 132 dots on a quadratic area and 32 dots on a rectangular area for the duration of 280 ms. From this table, we can obtain the diagram in Figure 14. With C we indicate the circular surfaces, with Q the quadratic one, with R the rectangular one.
Relations between shape (circular, quadratic, and rectangular), density of dots and average underestimation.
From these data, it appears that at equal densities, the underestimation caused by circular groups is approximately the same as that caused by quadratic groups, while the prevalence of one dimension is followed by a decrease in the underestimation or rather by a relative overestimation.
Figure 14 shows that, while the line that connects 1 with 2, 2 with 3, and 3 with 4 has an almost straight path, the line that connects 2 with 5, and 5 with 4 is broken. In short, the configuration with 132 elements arranged in a rectangle, having the shorter side as its base, is evaluated as richer in elements than the one composed of 142 dots, arranged on a circular surface. We assumed that the cause of this relative overestimation is to be found in the difference in balance between configuration 5 and the others.
We call this disturbance: the dimensional factor. We will specify the effect of this factor further down, let's say right now that this dimensional factor can be considered as an “overestimation factor” while the dimensional balance (of circular and quadratic surfaces) is an underestimation factor. As for the line connecting 1, 2, 3, and 4, it tells us that, when faced with circular and quadratic surfaces, the underestimation behaviour is equivalent and proportionally underestimated. In fact, the valid and expressive proportion between the given evaluations is given by: 72:32 = 142:64.
Before considering the data relative to particular groups distinguished from each other by differences in density or by formal accents (thus calling all the figurative or structural factors), we consider more closely the efficiency of the unidimensional factor: a factor that can say represents the first particular case of formal accent.
The data we have belong to various series of preliminary experiments in which the exposure lasted 100 ms. The number of subjects, among which the underestimators were predominant, was 17. The material to be evaluated was made up of groups of dots, homogeneously arranged on surfaces or along the sides of rectangular figures: the relationship between the sides, setting as a basis the smaller one, was 1:2, 1:4, and 1:8. To serve as a term of comparison, this material was contrasted with groups homogeneously arranged on roughly circular surfaces.
In one of these series, the subjects were presented with groups of 55 and 132 dots respectively, arranged on a circular surface (a), a rectangular surface (b: side ratio 1:2), a rectangular surface (c: side ratio 1:4); obtaining the average ratings given in Table 9.
From these data, it appears that the maximum dimensional balance corresponds to a maximum underestimation and that the one-dimensional factor (formal rectangular element) determines a certain relative overestimation which increases with the increase of the “rectangular character” (from b to c).
However, if this accentuation goes too far, beyond a deadline not yet well established, the relative overvaluation decreases.
Indeed, in another series of experiments, it was found that a group of 98 dots is evaluated on average (by the same subjects), when the surface was rectangular with a ratio between the sides of 1:4, as 64 dots, but of 54 when that ratio is 1:8.
This relative underestimation can be such as to exceed that due to dimensional balance: the subjects considered, in fact, evaluate a group of 98 dots, arranged on a circular surface, as 58 elements on average. Generally, it seems that the same laws also apply to groups of dots arranged along the sides of rectangular figures.
(c) Density. I report some experiments relating to square groups. Wanting to analyse the behaviour of subjects faced with the density of numerically constant groups, it is necessary to vary the area.
It is necessary for the surface to have its own constant shape. The formal structure chosen was the square; the objective groups are therefore dimensionally balanced. However, these do not correspond to dimensionally balanced subjective groups. Just as the vertical is overestimated compared to the horizontal, so too an objectively balanced group is subjectively transformed into a group with a rectangular character. Actually, each square subjectively appears as a barely visible rectangle in which the horizontal, subjectively shorter than the vertical, is considered the base.
In this series of experiments, four slides were used in which the number of dots was constant (= 98), the arrangement was homogeneous on a square surface, but the extension of the surface varied in an increasing direction from slide 1 to slide 4. Therefore, the density of the dots on the surface decreased (Figure 15).
Two squared configurations with 98 dots of different density. The figure shows only the first (n. 1) and last (n. 4) slide.
The projection of the slides was arranged in two ways:
Situation: small area: so that the dots of slide 1 occupied an area of 20 × 20 cm on the screen (minimum area) and the dots of the slides 2, 3, and 4 areas proportional to this.
Situation: large area: so that the dots on slide 4 occupied an area of 150 × 150 cm on the screen (maximum area) and the dots of slides 3, 2, 1, areas proportional to this.
The number of exposures, alternating with others of different types and times, was therefore 8: the times were always 100 ms; there were 17 subjects. The average ratings obtained are shown in Table 10 and in the diagram presented in Figure 16.
Average estimations as a function of both small versus large area and low versus high perceived density of dots.
From these averages we can draw two clear laws:
for small areas the density causes underestimation, and rarefaction (relative) overestimation
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; for large areas, density causes overestimation, and rarefaction (relative) underestimation.
In other words, making the elements rarer on a small surface increases subjectively their number, making the elements rarer, dispersing them, on a large surface, decreases subjectively their number.
Another fact is of particular interest, namely: the increase in overestimation, due to the greater density of the elements, with a large area, is almost equal to the overestimation given by the rarefying of the dots when the area is small. In fact, by collecting in a single progression the 8 values expressed in the sections broken lines in Figure 16, we see that the evaluation rises as the dots in the small area become rarer, and that it rises in the same measure as the dots in the large area condense. In fact, the line that joins, in Figure 17, the 8 values we have considered is almost straight.
Average evaluations rising as a function of increase in perceived dots rarefaction in small areas and of dots density in large areas.
The following proportion can be established even if with great approximation:
While reserving for another occasion to extend this research, I now report some data taken from observations made with four configurations. Of these, 2 had 62 dots (small dots in one, and large in the other); 2 had 102 dots (small dots in one, and large in the other). We collect in Table 11 the averages of the evaluations obtained with 17 subjects.
From these data, it appears that, with small dots, underestimation is a constant, as can be seen from the ratios: (d) The formal factor. At this point, I report the results obtained with a series of 8 configurations in which the number of elements was constant, while the variables were the architectural order of the groups, area and duration. The number of dots was 98. The configurations are those of Figures 18 and 19.
Four configurations with 98 dots arranged with different form and area (n. 1, 2, 3, and 4).
Four configurations with 98 dots arranged with different form and area (n. 5, 6, 7, and 8).
Below I collect the average evaluations of four subjects obtained with these stimuli. The stimuli were projected onto large surfaces. For configuration n. 8 in Figure 18 the size was 40 × 40 cm, and the other had sizes proportional to this. Duration was 1680 ms.
I sort them by increasing ratings from 1 to 8; numbers corresponding to those in Figures 18 and 19: 1 = 50, 2 = 52.5, 3 = 70, 4 = 73.5, 5 = 77.5, 6 = 84.7, 7 = 105, 8 = 122.5.
From these data, we have the diagram in Figure 20.
Evaluations for configurations 1 to 8 from Figures 18 and 19.
Now examining the groups exposed we see that the two maximums of undervaluation (group 1) and of overvaluation (group 8), correspond, the first to a configuration devoid of any formal accent, the last to that of the configurations which presents the maximum formal accent formal.
It does not surprise us that the degree of underestimation (50 compared to 98 elements) exceeds that of overestimation (122.5, again compared to 98), since, of the subjects considered, 3 out of 4 were of the underestimator type. In this case, therefore, a strictly quantitative comparison with the previously reported evaluations cannot be made, but rather a qualitative comparison.
A slight hint of a formal accent and dimensional character (n. 2) is enough to determine a certain relative overvaluation, and a fractionation, which accompanies and accentuates this character, gives us in n. 3 a new increase in overvaluation.
This element is of particular importance for the analysis of “type”; those who tend to experience a group as made up of subgroups will tend to overestimate it, while those who tend to grasp it as a single group will underestimate it. 18
The rarefaction of dots gives in n. 4 a new element of relative overvaluation. This result corresponds to what we saw above. The stimuli n. 5 and n. 6 made up of groups of dots correspond to averages of higher overestimation, so much so that we can distinguish the “nuclear or core factor” as an overestimation factor, for which the configuration appears to be made up of groups of dots, gathered around certain and distinct nuclei. If a given stimulus subjectively determines the impression of nuclei even if they are not objectively given (i.e., if the configuration is homogeneous), in all probability, it will provoke in the subject, depending on the density of the dots and the number of nuclei, a tendency to overestimate, the more so the greater the number of partial groups with which the configuration is experienced.
We can now notice, observing the configuration n. 6 of Figure 19 the new emphasis of the one-dimensional accent. While in n. 5 the overestimation given by the distinction of nuclei was contrasted with an underestimation factor given by the dimensional balance, in n. 6 where the area is given by a rectangle and that factor disappears, and there is a slight overestimation compared to the group n. 5.
The overestimation of configuration n. 7 may appear inexplicable based on the results achieved so far. Group n. 7 is in fact apparently similar to group n. 2. But if you look more carefully you can see how the dots, which in n. 2 are randomly scattered on a rectangular surface, in n. 7 are almost all aligned on horizontal lines parallel to the shorter side. This arrangement favours a new factor of overestimation: “the structural factor,” which encompasses both the unidimensional factor and the nuclear or core one which we have just mentioned.
Now taking into account that “figurative spontaneity,” which leads us to look for images in clouds, in spots, in many amorphous things, etc., and remembering how this is exacerbated when it comes to tachistoscopic perceptions we can conclude, summarizing what has been said so far, that: groups with a structure that is not clearly delineated are seen in an architecture that is subjectively much clearer than what the objective data justifies. 19
The evaluative behaviour of the subjects when faced with group n. 8 is clear. Here, the “multidimensional accent” element is added to the unidimensional accent and the action of what can be considered as a new factor of overestimation: the impression of vastness, determined by an empty surface, free of dots, and of great attentional emphasis.
The “large area” element surrounded by dots can only encourage overestimation. The same factors act in the groups of Figures 2 and 3 (see Introduction).
The large relative overestimation noted in the experiments with summarized in Figure 5 and 6, after these observations of ours, can be reduced to factors that can be determined separately. As in the experiments in Figures 18 and 19, a minimum rating of 50 is contrasted with a maximum rating of 122.3, as shown in Figure 5 (objective number = 69) a minimum rating of 42 is contrasted with a maximum of 80, and the two groups in Figure 6 (objective number = 90), give a minimum evaluation of 50 dots, and a maximum of 98. It is worth noting that, in all these comparisons, we find minima and maxima of evaluations that are related to each other approximately as 1 is to 2.
The numerical data are:
42 = 80 (Figure 5), 50 = 98 (Figure 6), 50 = 122.5 (Figures 18 and 19).
In the third group where the greater the difference, the greater the gap between the over- and underestimation factors.
Before moving on to the consideration of groups of dots in movement, I report the data of some experiments made with the material in Figures 18 and 19, experiences which differ from the previous ones because the area was very small (10 × 10 for group 8) and the exposure lasted 280 ms instead of 1680 ms. This experiment had to serve as a counterproof to the previous one, as the very small area and the very short exposure greatly reduce the clear figural perception and take away prominence from those elements that we have seen as factors of overestimation. The average ratings obtained by subjects are as follows:
However, while in the previous series, the overestimations progressively increase from n. 1 to n. 8, in the small area and exposure series there is a great uniformity of behaviour. The groups n. 3, n. 1, and n. 7 on the one hand, the groups n. 2, n. 5, n. 6, and n. 4 on the other oscillate around the values of 35 and 45 units, while the group n. 8 alone gives us a relative overestimation of 72, and it is natural that it is so because this is the only case in which the structural element is so evident, despite the smallness of the area and the brevity of the time, that it clearly affects the evaluation. In all other cases, the brevity of exposure and the smallness of the area prevent the clear perception of structural elements and standardize the evaluation, bringing us back to the underestimation characteristic of small areas with homogeneous groups.
Thus clarified the importance of some factors that act in the face of “static” groups of dots and precisely the factors:
of vividness or perceptive liveliness; of density; of the exposure duration; of the size of the area; of the contour and surface; of the one-dimensional accent; of the formal and nuclear factor; of the impression of vastness;
Let us now consider evaluative behaviour in the face of “dynamic” stimuli (moving dots).
Before moving on to these new experiments, it seems useful to me to report just one piece of data taken from the introspective protocols drawn up after the immediate evaluation. From the protocols, it appears that the assessments appear introspectively more reliable to the subject, more certain, when the area is small. Now, precisely in this case, we can see how they are instead less adequate. In the present instance, the impressions that subjectively satisfy the most may be less exact: probably what gives the sense of being closer to reality, that is, the small area gives a sense that the evaluation is more exact, and an impression of “stability,” of unity.
And this is a theoretically significant result.
The Moving Dots
Amorphous groups. The chymoscopic factor. The groups of experiments that will now be described were carried out with the device described in paragraph The devices. Let us remember that the times taken by individual groups of dots to pass through the visible field (slit) were:
2600 ms: fast movement (v) 4600 ms: medium movement (m) 6500 ms: slow movement (l)
A first series of experiments was carried out with amorphous groups arranged on a constant surface; they passed through a large slit (height 40 cm). The average ratings obtained with five subjects are collected in Table 12; we report these data in the graph of Figure 21.
Trend of average evaluations of amorphous groups moving with a slow, medium, or fast speed.
The average scores are reported for all subjects regardless of “type.” If, however, we take distinct account of the values relating to the subjects who, in experiences with static communities, proved to be over- or underestimated, we see that their “type” remains constant even in these experiences.
We report in Table 13 and in the diagram in Figure 22, by way of example, for the maximum and minimum speed, the data relating to two subjects, one belonging to the underestimator type and the other to the overestimator.
Trend of average evaluations of amorphous groups moving with a slow or fast speed made by one underestimator and one overestimator type.
The difference in the results is clear, corresponding to the different speed of movement.
The fast movement favours the overestimation of groups up to 120 units, while the slow movement, and even more so the medium one, favours an underestimation of groups exceeding 50 units.
For large groups, therefore, the movement generally favours underestimation. However, it is good to take into account the fact that it is a special movement: the one in which you see the elements “passing by,” as if you were seeing an object passing in front of a window while keeping yourself a few meters away.
Another non-negligible factor is what could be called chymoscopic: due to its action we see the dots pass, not uniformly through the slit, but in “waves” or “swarms.”
In this regard, by way of example, I report some descriptions taken from the introspective protocols. Subject C.M. says: “the dots I saw appeared in waves. There must have been four or five waves.” Subject C.R.: “the dots were in waves, ever closer and almost pressing…” Subject E.G.: “the stitches went in jerks as if someone was pushing them… they went on in packs like frightened beasts.” The subject M.S.: “dots pass as masses.” Subject N.P.: “I was unable to count the waves, it seemed to me that the rhythm of the device corresponded to the waves of dots.” The action of the chymoscopic factor can perhaps be considered corresponding to that of the nuclear and structural factors, which cause an overestimation.
As for the relationship between movement speed and evaluation, there is something else to observe: from the data collected in Figure 21 we see that the maximum underestimation occurs when the speed of movement is average, while both an increase and a decrease in speed determine a relative overestimation. We are thus faced with processes similar to those noted regarding the duration of exposure.
(b) Formal groups in movement: 1. Tachykinetic experiments on a large area. Let's keep in mind the material described in Figures 18 and 19: these configurations were exposed in fast movement through the same slit as in previous studies (large area): their structure, perfectly evident in the static case (considered in paragraph Grouping and objective arrangement, (d) The formal factor) could not be experienced in these kinetic experiences, if not partially and with particular subjective deformations (for example that given by the chymoscopic factor).
In Table 14 we report the average ratings of the present experiments in relation to those of the static series. We ordered the groups (indicated with the same numbers as in Figures 18 and 19) according to the progression of the evaluation. The kinetic results are very different from those obtained in static conditions.
The strongest discordances are found in n. 5, n. 1, and n. 2; where the evaluations go, in n. 5, from a relative overestimation (static series), to an absolute underestimation (kinetic series); in 1 from an absolute underestimation (static series), to a relative overestimation (kinetic series); in 2 from an absolute underestimation (static series) to a pure overestimation (kinetic series) (Table 15).
Let us consider the cases relating to groups n. 1 and n. 2 (see Figure 18) which in the static series gave almost equal underestimations and here give respectively a relative and an absolute overestimation. The reason can be found in the different function that density has in the two cases of static and kinetic experiences. As we saw above, density is, for large areas, a factor of underestimation, when, however, it does not give rise to the onset of grouping factors, as was the case with group n. 7 in Figure 19. In the present experiences the density favours an impression of a “wave” succession of the perceived elements, that is, the onset of that chymoscopic factor, which, as we have said, is a factor of overestimation.
That grouping n. 5 now gives rise to a maximum underestimation, in tachykinetic experiences, is understandable, given that, due to its particular configuration, it hinders the onset of that impression of “waves of dots” which corresponds to compact groups: and this even without taking into account the absolute impression of poor, sparse dots that this group arouses. In fact, by extracting some observations from the protocols of these experiences we find that: the subject C.M. says: “this time the impression of ‘waves’ is as if detached from the groupings of dots…”; the S.D.M. subject: “the dots seemed ‘few’ to me, they were more sparse than in previous experiences”: the M.S. subject: “I see that the dots are very sparse and therefore very few…”
The three cases of maximum difference can be said to be clarified.
The results obtained with groups n. 7 and n. 8 agree with what we stated. Compactness and multidimensional accents act in a concordant manner: perhaps in n. 8 a double chymoscopic impression given by the two horizontals that cross the slit. We find the maximum overestimation for this group (135), a value that exceeds the number of exposed dots by 37 (98).
2. Tachykinetic experiences on a small area. The values relating to the evaluations obtained with a very small slit (area) (12 cm in height) are contrasted below with those relating to the experiences with a large slit.
We show these values in Figure 23, where one line indicates the data relating to the large area, and the other the data relating to the small area.
Trend of average evaluations of groups 1 to 8 of Figures 18 and 19 obtained with a small versus large slit (area).
We observe first of all that, in complete agreement with what we gained from previous experiences with static series, the narrowness of the area is accompanied by underestimation; which in the present case is an absolute underestimation. None of the answers reaches 98 which is the objective number of dots.
This result is both relevant and reliable, since we are dealing with the same subjects. In only two cases the values of this series correspond to those of the large slit series; that is, the values relating to groups n. 5 and n. 4.
In this series, in which the movement of the dots is fast, and the slit is small, the formal elements of the groups are difficult to grasp: the influence of the formal factor therefore remains very limited. While the evaluation limits in the large slit series are between 35.5 and 135, in the small slit series they are between 30 and 85. While the maximum and minimum of the first series are in the ratio 1:37, those of the small area series are only in the ratio 1:2.4.
The Stability of the Factors
From all these observations it appears that the factors considered as external factors, as objective determinants of evaluative behaviour, are truly stable factors, capable of producing particular subjective perceptive behaviour, such as to determine evaluative oscillations uniform enough to be able to determine their laws. If these external factors act stably on the behaviour of the subjects, we can say, anticipating one of the fundamental results of the introspective series, that the internal factors, those that presumably constitute the over- or underestimating “type,” act stably in the same way: so it is clear from the introspective protocols that, alongside the evaluation not based on any rational element, not justifiable for the subject himself if not based on the pure absolute impression given by the group, another evaluation often arises that the subject obtains on the basis of rational elements. But to the subjects, however, the immediate evaluations, relating to the first impression that arose spontaneously, appear more reliable than the subsequent ones, due to mental processing of the first impression.
Thus, for example, if a first impression in front of group n. 8 in Figure 19 is 70 (or 150 depending on the type) and a reflected evaluation, based on the analysis of the perceived group, leads by reasoning “the dots will be 25 or 30 per side: 25 × 4 = 100,” to an evaluation of 100 instead of 70, the impression that imposes itself is not the latter, that is, the most reasonable one, but the former.
Thus, by way of example, the subject C.M. (underestimator), in front of an amorphous group in movement says: “despite the impression that the waves were 4 or 5 and the dots 25 per each wave, I would evaluate the whole as I evaluated it ‘at first sight,’ i.e., 75 dots, and this impression is the most stable…”
And S.D.M. (overestimator) says: “The number 162 which I knew was the objective number gave me a sense of unease at the thought, I thought of 175 without being satisfied, then of 200, and finally of 220.”
That is, in these processes of consciousness, those situations that we find in other cases are repeated: the first impression we have of a person, or an object remains, for example, more solid and certain than the one that subsequent reasoning and impressions try to build.
This fact cannot be surprising: both in the experience of ordinary life and in simple laboratory experiences, a given subject maintains its nature: the laws of the evaluative mechanism remain unchanged in the face of all objects.
Summary
We attempted to experimentally specify the main factors on which immediate numerical evaluations (i.e., independent of reasoning processes) of groups of dots presented with a tachistoscope depend. Having established that the subjects can be distinguished into two types, underestimator and overestimator, having analysed the absolute impressions of quantity, the influence that they exert on the results of the evaluations was mainly studied: the variation in the duration of exposure of the groups, in the size of the area in which they appear, etc. Situations in which the groups, rather than appearing immobile on a surface, are seen moving through a slit are considered separately.
Received at the Directorate Editorial Board on 8 June 1929 - a. VII.
Footnotes
Authors’ Note
In many places we translate “collectività” with “numerosity.”. “Numerosità” exists in Italian as a word, but it was not a term used by De Marchi. However, the English collectivity is not a good translation of the Italian “collettività.” A kymograph is a revolving drum wrapped in paper on which a stylus draws to record changes. Most participants in Benussi's laboratory were students and his two assistants (Bobbio & Roncato, 2023). Their initials can be identified, here C.M. probably stands for Cesare Musatti, and S.D.M. for Silvia De Marchi.
Acknowledgements
We are grateful to Sergio Cesare Masin for helpful comments on an earlier version of this article. The Authors wish to dedicate the paper to Riccardo Vittorio Elia Musatti (1933–2023), the only child of Silvia De Marchi and Cesare Musatti.
Author contribution(s)
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Fondazione Cassa di Risparmio di Padova e Rovigo, CARIPARO (grant number RSE 2021 project SHARE).
