Tag: Larry Hecht

By Larry Hecht

I can conceive in the mind of six objects, whose relationship to one another I wish to investigate. Their character as real objects does not interest me, but only that quality which makes them distinct, thinkable. They are, thus, objects in thought. I will label them with the number designations 1 to 6, though I might equally denote them by letters, or any other symbols which allowed me to keep them distinct in my mind. I am interested in discovering the number of different ways these six distinct objects can be formed into pairs. Their representation by numbers, allows a convenient means of investigating this. I first list all the pairs of 1 with the other 5, then all the pairs of 2, and so forth. The result is summarized in the table:

12

13 23

14 24 34

15 25 35 45

16 26 36 46 56

== == == == ==

5 4 3 2 1

Counting the number of pairs in each column and summing them, produces 5 + 4 + 3 + 2 + 1 = 15 pairs.

In another form of representation, I can imagine the six objects as points on a circle, and portray their pairing as the straight lines connecting any two. Drawing them produces a hexagon, and all the straight lines that may be drawn between its points. Counting all the connecting lines, we find 15, the same as the number of pairs above! The mind rejoices in the discovery of the equivalence of the two representations.

Closer examination of the second form of representation, now reveals also a difference with the first. In the first, nothing distinguished one pair from the next, except the symbols used to designate them. In the second, we discover three distinct species of relationships among pairs, each characterized by a different length of connecting line. We have (i) the six lines forming the sides of the hexagon; (ii) the six somewhat longer lines connecting every other vertex (i.e., 13, 24, etc.); (iii) the three longest lines connecting diametrically opposite vertices (14, 25, 36).

Where, before, the mind celebrated the sameness, it now rejoices at the difference of the two forms of representation, and is impelled to look for its cause. We hypothesize that the difference must reside in a property of the spatial mode of representation. We may reflect that, from the manifold ways we might have chosen to arrange our six points in space, we chose to place them on the circumference of a circle, equally spaced. An arbitrary arrangement of six points in a plane would have produced another, less-ordered relationship among the pairs. Another arangement, a spiral perhaps, would have produced a richer ordering.

Thus, from the positing of relationship among things in the mind, we moved to two modes of representation of that relationship, then to their sameness and difference, then to the causes of that difference. Having hypothesized that the latter is the result of the spatial form of representation, we are next led to explore the variety of such representations.

Of the great variety of possibilities, we choose now to rise above the plane, in order to examine the relationship among six points in three-dimensional space, the familiar backdrop for our visual imagination. Just as the circle aided us in ordering the points in the plane, here its counterpart, the sphere, comes to our aid. Six points, spaced evenly around the surface of a sphere, form the vertices of the Platonic solid known as the octahedron. We can picture two of its six points at the north and south poles of a globe, and four more forming a square inscribed in the circle of the equator. Connecting each point to its nearest neighbor, we find the 12 lines which form the 8 equilateral triangles, which are the octahedron’s faces. But we have not yet connected the six points in all the ways which space allows. Each point can yet be connected to its opposite, forming 3 more lines, which are diameters of the circumscribing sphere. Behold, again, the 15 paired relationships of six objects, now clothed in a new ordering, this time of two species!

We may now compare the three modes of representation our mind has invented to investigate these pairings:

1) By number, which produced the series 1 + 2 + 3 + 4 + 5 = 15.

2) In planar space, using the circle, which produced the three species of lines connecting the points of the hexagon.

3) In space, using the sphere, which produced the two species of lines connecting the vertices of the octahedron.

In turn, each of these modes of representation suggests new investigations. For example, with respect to the first (i.e., number), we may inquire into the pairwise combinations of other numbers of things, from which we soon discover that, in general, for “n” things, the number of pairs that can be formed is equal to n(n-1)/2, and we may next inquire, what is the expression for combinations three-wise, four-wise, or n-wise?

With respect to the second (the distribution of points on a circle and their combinations), we discover that there exist species beyond the regular polygons, which are known as the star (or Poinsot) polygons. These cannot be generated out of any arbitrary number of points, but only when the number of points, and the order in which we take them, are relatively prime to each other (that is, have no common divisor). The first of the star, or Poinsot, polygons, appears when we take five points on a circle, and connect every second one until the figure closes (that is, 1 to 3, 3 to 5, 5 to 2, 2 to 4, and 4 to 1). The result is the star pentagon, or pentagram, which is conveniently described as 5/2. We can then discover the 7/2 and 7/3, the 8/3, the 9/2 and 9/4, and so forth.

With respect to our third mode of representation of the pairwise combination of things (the distribution of points on a sphere), a new ordering principle arises: that a perfectly even distribution is only possible in the cases of 4, 6, 8, 12, and 20 points. When we investigate these, we find species of pairwise combinations called edges, diagonals, diameters, and some others, the greatest variet of species occurring in the 20-point figure.

Now, let us reflect on the higher ordering principle: All of the representations we have given, even the spatial, are creations of mind, products of the arithmetic or visual imagination. Yet, so real do these creations of the mind seem to us, we may be tempted to marvel at them as if they had some existence outside of the mind. (“But Platonic solids are {real}. I can build them!” you say. Perhaps you never have. Anyone who has tried, soon discovers a, sometimes gooey, massiness where massless points are supposed to be, a very finite thickness to the infinitely thin lines of the edges, and a, sometimes wrinkly, bulk to the massless surfaces. Even three-dimensional space, the forgiving medium of all our constructions, which seems so certain, so real, is only the ingenious work of the mind, the visual imagination. All are products of the mind.)

But when, in nature, the mind discovers forms just like these we have just created (thought), put there not by us, but by something like to us in mind, yet much vaster, then may we truly marvel, and reflect: What makes nature makes us. What we make in mind, think, is then nature — and may be so in a higher form than what we perceive outside us. (The proof of this truth, well-known to readers of this publication, need not be repeated here.) So in the ordering, number, space, and mind, the mind stands at both ends of the series, as both creator of its own images, and perceiver of others; the one is called imagination, the other, reality. Yet they are both real, as we just showed, and even both imagined, in so far as the perceived external is {known} only through the images of mind.

With such considerations, true science begins.

Larry Hecht

Diophantus, who lived probably around 250 A.D., wrote a book called {On Polygonal Numbers,} of which only fragments remain. One of the famous fragments refers to his work on a definition by Hypsicles, an earlier Greek mathematician, concerning polygonal numbers. Working out what Diophantus means in this short fragment proves quite interesting, and relevant to the topics we have been discussing in these series. I will first give you a translation of the fragment from Diophantus. Don’t worry if it seems incomprehensible at first. We will construct it, and then it will all be quite clear.

Diophantus writes:

“There has also been proved what was stated by Hypsicles in a definition, namely, that `if there be as many numbers as we please, beginning from 1 and increasing by the same common difference, then, when the common difference is 1, the sum of all the numbers is a triangular number; when 2, a square number; when 3, a pentagonal number. The number of angles is called after the number which exceeds the common difference by 2, and the sides after the number of terms including 1.'”

To understand what he means, let’s take the most familiar case, that of the square numbers. Most books discussing this subject (sometimes referred to as the “figurate numbers”) draw dots as illustration; but there is a flaw in this, which you will understand after we have done the complete construction. It is far better to find some square objects, or cut them out of paper. Using these square tiles as the units, you will discover that only certain numbers of tiles go together into squares. The first grouping is 1, the second 4, and the third 9. But you should construct this for yourself, for it is already telling you something important about a certain kind of bounding condition, which interested Kepler very much.

Now, cut out some equilateral triangles and do the same thing–that is, make triangular numbers. This is a little less familiar, so I will illustrate how to count in triangles for you:

    /\     1                  /\/\/\/\    4

	/\/\    2                  /\/\/\/\/\  5

	 /\                          /\
	/\/\    3 (2-triangled)     /\/\       6 (3-triangled)
	                           /\/\/\        

You see that the first three triangular numbers are 1, 3, and 6. These have sides of lengths 1, 2, and 3, just as the first three square numbers (1,4,9) do. You might notice that there are also holes in these numbers, which the squares did not have. There is no need to worry about them. You will see by the end, why they must be there.

Finally, we come to the pentagonal numbers. Now, you must cut out at least 5 equal pentagons, although 12 would be better. Here the fun began for me: to figure out what 2-pentagoned would look like. As I don’t want to spoil it for you, I will not say right here, but let you pause and puzzle over the construction a bit. For now, I will give you the numerical values: the first three pentagonal numbers are 1, 5, and 12.

Now, it is easy to see from these constructions what Hypsicles had discovered, and described in words. We can illustrate it in the following series:

Triangular numbers (common difference = 1)
	Series: 1  2  3  4   5   ...
	Sums:      3  6  10  15  ...

Square numbers     (common difference = 2)
	Series: 1  3  5  7   9
	Sums:      4  9  16  25

Pentagonal numbers (common difference = 3)
	Series: 1  4  7  10  13
	Sums:      5  12 22  35

In each case. we start with one, and increase by the common difference, characteristic for the series. The sum of the numbers in the series is the number of tiles we had to employ to make the triangular, square, or pentagonal numbers.

This may all seem innocent enough, but there is a “fighting” matter of epistemology buried within. It is the same point which Gauss addresses from a more advanced standpoint, in his refutation of Euler, Lagrange and d’Alembert’s attempts to prove the Fundamental Theorem. Namely, do we accept any notion of number, or operations on number, that is not constructible, or subject to “constructible representation” (as Gauss once described the same issue respecting a matter in physics)? It is not only a fighting matter for us. Our enemies also get very upset over the issue. I recognized how much so, after I contemplated why the translator of the Loeb Classical Library Edition {Greek Mathematical Works, II} felt it necessary to add the bracketed phrase “[; and so on]” following the words “when 3, a pentagonal number” in the citation from Diophantus that I gave above. If Hypsicles or Diophantus had wished to say “and so on,” why would they not have done so? Sir Thomas Heath, the leading British commentator on these matters, finds it a shortcoming that Hypsicles had not gone further than the pentagonal number, and claims that what Hypsicles was really showing was how the n-th term of a series, with any common difference, could be determined.

Yet, anyone who has properly considered the significance of the Platonic solids, and stuck to the principle of mathematical rigor employed by both of Gauss and his Greek predecessors, would immediately recognize why Hypsicles stopped at the pentagon. What is being considered is not a math-class game of number series, which seem to go on forever to a bad infinity, but a process of examining the lawful constructibility of number. There is a clue to this also in the {Theaetetus} dialogue of Plato, which had been in the back of my mind, as I wondered what was getting Heath and company so worked up. Consider how Theaetetus describes there, in his examination of the problem of incommensurable numbers, the generation of the numbers 1, 2, 3 as the sides of the square numbers 1,4,9. He calls the numbers 1,2,3 “powers” (where we were taught to call them “roots”), because they have the “power” to generate squares, the singularity under examination in this case. The point in both cases, is that number must be lawfully constructed, and it is obvious that Hypsicles was doing so by examining the paradoxes generated by the Platonic solids.

So, let us now see what happens, if we take these polygonal numbers into the next dimension. The case most familiar to us is that of the square turning into a cube. Thus 1-cubed is 1, 2-cubed is 8, and 3-cubed is 27. (Remember, we are not doing a multiplication table operation, but a construction.) What, then, is the equivalent construction for the other polygons? We can see the case for the triangle most easily, if we now build ourselves four tetrahedra (that is, the Platonic solid made of four equilateral triangles), using triangles of the same size as those we cut out for the construction of the triangular numbers. Construct again the triangular number three, and place a tetrahedron atop each of those triangles. Then, place one more tetrahedron at the summit. Examining the solid so constructed, you will see that it has sides of length 2 in every direction–hence we have constructed 2-tetrahedroned. You can figure out for yourself, what 3-tetrahedroned would be [; and so on].

You might have noticed that there was a hole on the inside of the figure 2-tetrahedroned. That space in there turns out to be an octahedron, by the way. We also had those holes in the plane when we built the triangular numbers. This is telling us something interesting about the tiling of the plane, and the filling of space. Only squares and hexagons, among the regular polygons, can tile the plane, and only cubes and rhombic dodecahedra, among the regular (or quasi-regular) solids can fill space without gaps, which you can investigate for yourself, as Kepler did to his great delight. If you try to tile the plane with pentagons, you notice that when three come together at a point, there is an overlap. That is the key to constructing the figure 2-pentagoned, which I left for you to figure out earlier. You must break the unwritten rule in your mind, and allow yourself to overlap the sides.

Now, if tetrhahedrons do not quite fill space, but leave gaps, and cubes just manage to fill it up, you might expect that dodecahedra would go too far. and overfill it, just as the pentagons overtiled the plane. If you have now constructed your 2-pentagoned figure, with the overlapped sides, you can try your luck at placing a dodecahedron atop each of the five overlapped pentagons, and another dodecahedron atop each of these, to produce the number 2-dodecahedroned. You will see that, just as the pentagons had to overlap, so the dodecahedra must overlap, or interpenetrate, and so the figure 2-dodecahedroned will be of a different type than the cubic or tetrahedral numbers.

Those of you who know why there cannot be more than five regular solids, will now see why Hypsicles stopped at the pentagon. For, while the series with increasing common differences can be extended out to a bad and boring infinity, the interesting paradoxes are not going to arise, unless we have a concept of a constructive process for these numbers.*

Before closing, and since you have all the materials at hand, let us review why there can be only five regular solids. It is a famous proof, given by Kepler. The regular solids are, in fact, the plane projections of the figures produced by tiling the sphere, and they are five ways to do it. As plane solids, they must have regular polygons for their faces, so the problem is reduced to great simplicity by considering only how many of these figures may come together at a vertex. Start with the equilateral triangle. Three of these may be joined at a point, and brought together into a sort of cup, so that they could hold water. This will become the vertex of the tetrahedron. Four triangles may also be brought together, and cupped; they will form the vertex of the octahedron, which looks like two Egyptian pyramids brought base to base. Five triangles may also be brought together and cupped; they form the vertex of the 20-sided icosahedron. However, when six triangles are brought together, it is seen that they just lie flat, and cannot be made into a vertex of anything solid. Next, we try the square, and find that three can be brought together and cupped into a vertex of what becomes the cube. But four are too many; they lie flat. Three pentagons lying in the plane, and joined at a vertex, leave just enough space to be cupped into a vertex of what becomes the dodecahedron. But that is the end of the possibilities, for if we next take a regular hexagon, we find that when three are brought together at a point, they simply lie flat and cannot become the vertex of any solid.

So, you see, there is no “[and so on],” as Hypsicles and Diophantus appear to have understood better than their modern commentators of Oxford erudition. Avoiding “[and so on]” is also good advice for your speaking practice–that you not recite a series of things in sing-song fashion, as so many people do these days, as if there were no lawful cause for their being there. This is nominalism in language, as the idea of number without constructibility is nominalism in mathematics. It is part of the same disease, which we are trying to cure.

—————————–

* Of such interesting paradoxes, you might consider, as a topic for more advanced consideration, that a prime number is a constructible species in the series of numbers constructed using squares as the tiles. Following Theaetetus’s specification that we allow only square or oblong (rectangular) numbers, the prime is a number of the form that it can only be represented as a rectangle of width 1. What, then, is a prime number in the triangular or pentagonal series? What else is peculiar about the square and rectangular numbers?

Pierre de Fermat became quite fascinated with the polygonal numbers, and discovered many things about their properties of combination. His famous Last Theorem might be seen as an investigation of the constuctible properties of solid numbers of the square, cubic, and higher variety.

by Larry Hecht

The evident success of the ongoing project to measure the retrograde motion of Mars, suggested to me that we are ready to take up another challenge in observational astronomy–the construction of a solar astronomical calendar.

This is a challenge that Lyn posed to us nearly 25 years ago, in part influenced by a trip to India, where he came into contact with the work of turn-of-the-century Indian independence leader Bal Gandaghar Tilak. I first began to seriously take up Lyn’s challenge in connection with my own efforts to understand Tilak’s work some time in the mid-1980s. To be honest, I could not understand at first why Lyn kept talking about “constructing a calendar,” which I thought was something easily obtainable at any stationery store. Once I began to understand what was involved, however, I found that this project led in a number of very interesting directions.

Tilak’s work involves the hypothesis that verses in the sacred Vedic hymns refer to astronomical phenomena, which could only be known by a people living at a point at or above the Arctic Circle. His hypothesis immediately brings into play at least three important and interlocking branches of science: astronomy, Indo-European philology, and climatology, all necessarily subsumed under the topics physical economy and universal history. One of the most provocative aspects of Lyn’s discussion on the subject was the hypothesis that a highly-developed, poetical-musical language, (such as was indicated by the Sanskrit, for example) would be required for the task of recording and preserving astronomical observations over long periods of human history. Rather than the object-fixated grunts of some doomed society of primitive Rave-dancers, forms of verbal action capable of expressing the transformative nature of natural law would be required, including verbal forms capable of expressing the subjunctive mood necessary for any hypothesis, varying degrees of completion of action, and many other subtleties.

Attempting to read Tilak’s {Arctic Home in the Vedas,} however, produced some immediate problems. Early in the book, the author began talking about astronomical phenomenon, such as the precession of the equinox, the seasonal motions of the Sun, and the relationship of the Sun to the zodiacal constellations, which, I soon realized, I had no real understanding of. To make any sense out of his thesis, it was clearly necessary to have some grasp of these things, so I decided at some point to dig in and make the effort. As I had been reading books about astronomy since childhood, it was something of an embarassment to have to admit to myself that I could not even explain the meaning of the seasons in any cogent way. A joke which Jonathan T. had been making at that time had stuck in my mind, and was helpful in overcoming the embarassment. The joke, which I think he may have included in the title of a Fusion magazine article he wrote at the time, was the phrase “Astronomy without a Telescope.”

A Simple Calendar Observatory

The method I suggest here for constructing a solar astronomical calendar, is not an exact replica of the steps I took. However, I think it will work, and, under the present circumstances, where collective and enthusiastic pedagogical activity is taking place all around, it should allow us to proceed quickly and happily.

I suggest we begin by constructing something which will resemble, in principle, the famous Stonehenge, an historical artifact which has unfortunately taken on all sorts of cult-like significance, but which is actually just one of many still-standing astronomical observatories from the Megalithic period. Our observatory will be much simpler. Probably the most difficult part of this project will be to find a level site with a good view of the horizon, especially to the east and west, to which we can return regularly. The calendar observatory need consist of no more than some stakes in the ground, arranged around part of the circumference of a circle, and one stake at the center.

Now, here is what I suggest we do. On the first day, we make two observations, one at sunrise and one at sunset. We begin by locating a center for our circle, and driving a stake in the ground at that point. This will be the siting post for all the observations. Now, choosing an appropriate circumference for our circle, and using a rope or chain to keep a constant distance, we plant a stake in a line from the siting post to the point where the Sun rises over the horizon. We return before sunset, and similarly drive a stake in the ground on the other side of the circle where we see the sun set.

That simple observation, repeated over the course of a year, will provide us with an experimental understanding of many important concepts in astronomy, including the summer and winter solstice, the vernal and autumnal equinox, and the equation of time (which, by the way, bears a certain relationship to the lemniscate). But this is only a beginning. For, using no more than our simple observatory, we may next begin to observe the motion of the Sun, not only with respect to fixed positions on the ground, but also with respect to the stars. From this we may develop many new concepts, including that of the precession of the equinox, which plays an interesting part in this history of science, which, we shall also come to see, is the history of language.

But we will also have an advantage over our predecessors, who were carrying out such observations probably tens of thousands of years ago, several cycles of glaciation back into the pre-historic past. By use of modern means of communication, we will be able to rapidly compare observations made at widely divergent positions on the Earth. We shall have the great advantage of having access to observations at the high northern latitudes of Stockholm and Copenhagen, the near-equatorial latitudes of Bogota and Lima, and many middle latitude sites in both the Northern and Southern Hemispheres. This will really make for some fun, some paradoxes, and definitely ensure that there is no “right answer” to be looked up in the back of the back.

To start out, I suggest we take the time to explore and secure a good site for our calendar observatory, and begin with the first very simple observation of marking the rising and setting points of the Sun. Between these points, we will have a circular arc on the ground, whose angle can be measured and recorded. It would also be useful to make some observations of the path of the Sun in the sky over the course of a day. From this observation and the position of our two stakes in the ground, we should also be able to come to a clear understanding of the meaning of North, South, East and West, and also of the word Noon. For some added fun, we might try to measure the highest declination of the Sun, and observe what time it occurs on our watches.

With the measure of the circular arc between the two stakes in the ground recorded, it will be most interesting to immediately compare the results with those found on approximately the same day at other calendar observatories around the globe, as one could do, for example, on an international youth call. If it should happen that some of the observations should take place around the 22nd of September, a very interesting paradox will arise when the observations from different latitudes are compared. (The path of the Sun and its position at Noon ought also to be observed on that day.) But it will only get more interesting, as the subsequent observations are taken, and compared for the different latitudes.

So, let the fun begin.

Larry Hecht April 21, 2006

[Figures for this pedagogical can be accessed at: www.wlym.com/~bruce/periodic.zip]

Dmitri Mendeleyev discovered the concept of the periodicity of the elements in 1869 while he was in the midst of writing a textbook on inorganic chemistry. The crucial new idea, as he describes it, was that when the elements are arranged in ascending order of their atomic weights, rather than simply increasing in some power or quality, he found periodically recurring properties. Mendeleyev noted explicitly that this discovery led to a conception of mass quite different from that in the physics of Galileo and Newton, where mass is considered merely a scalar property (such as F = ma). Mendeleyev believed that a new understanding of physics would come out of his chemical discovery. It did, in part, in the developments that led into the mastery of nuclear processes, even if the flawed foundations of the anti-Leibnizian conceptions injected by British imperial hegemony were never fully remedied. The development of the sort of conception connected with Dr. Robert Moon’s nuclear model will help to fulfill Mendeleyev’s insight on this account.

There are just 92 naturally occurring elements in the universe. Their existence and organization in the periodic table discovered by Mendeleyev is the most fundamental fact of modern physical science. We will soon see how the discovery of radioactivity and nuclear power, among so many other things, would not have been possible without the prior achievement of Mendeleyev. Let us first get a general idea of what the periodic table is, and then examine some of the considerations which led Mendeleyev to his formulation.

The periodic table systematizes the 92 elements in several ways (Figure 1). The horizontal rows are known as {periods} or {series}), and the vertical columns as {groups}. The simplest of the organizing principles is that the properties of the elements in a group are similar. Among the many properties which elements in a group share: Their crystals, and the crystals of the compounds which they form with like substances, usually have similar shapes. Elements in the same group tend to combine with similar substances, and do so in the same proportions. Their compounds then often have similar properties. Thus sodium chloride (NaCl) which is table salt and potassium chloride (KCl) combine in the same 1:1 proportion, and show similar chemical and physical properties. Partly because they tend to make the same chemical combinations, the members of a group and sometimes adjacent groups, are often found together in ore deposits in the Earth. For example, copper usualy occurs in ores with zinc and lead, or with nickel and traces of platinum. If you look at a periodic table, you will see these elements in nearby adjacent columns. Or for another example, when lead is smelted, trace amounts of copper, silver, and gold (which occupy a nearby column to the left), and arsenic (in the adjacent column to the right) are found. We will look at more of these sorts of relationships shortly.

(To prevent confusion, we should interject this note of warning. When the periodic table is taught in the schools today, it is usually presented as an ordering principle for the electron shells which are thought to surround the nuclei of atoms. The modern explanation of chemical reactions invokes the interaction of the outer electrons in these shells. It is important to understand that at the time of Mendeleyev’s discovery, no chemist had any idea of the existence of an atomic nucleus nor electrons. The electron was considered as a theoretical entity in the electrodynamic work of Wilhelm Weber (1804-1891), but this had little to do with chemical thinking at the time. The first approximate measure of the mass of the electron came in the first decade of the 20th century, and the validation of its wave properties came in 1926. In the prevailing view of the atom at the opening of the 20th century, there was no central nucleus, but rather a homogeneous spread of charges. Thus, to understand how Mendeleyev came to his discovery of the periodic table in 1869, we must discard most of what we might have learned of the subject from modern textbooks. If we feel a slight pang of remorse in giving up what little we think we know of the subject, we shall soon find that we are rewarded by a far greater pleasure in discovering how these discoveries really came about. We shall then also be at the great advantage of knowing where the assumptions lie which will surely need correcting to meet the challenges of Earth’s next 50 years.)

By arranging the elements in increasing order of their atomic weights, Mendeleyev found that they fell into periods which repeated themselves in such a way that elements possessing analogous properties would fall into columns one below the other. Within the periods, many properties, including the valences (defining the small whole number proportions in which the elements combine with each other), the melting and boiling points, and the atomic volumes (which we shall discuss further on) showed a progessive increase and decrease which was analogous for each period.

By examining these periodic properties, it was also possible to see that there were gaps in the table. Some viewed those gaps as a weakness in Mendeleyev’s hypothesis. But Mendeleyev was convinced the conception was right, and that the gaps represented elements still to be discovered. He worked out the probable properties of some of these unknown elements on the basis of their analogy to the surrounding elements. Within a few decades of Mendeleyev’s publication of his periodic concept, several of these missing elements were discovered.

For example, in the Fourth Group (the 14th in the enlarged numbering system adopted in 1984), below the column containing carbon and silicon, Mendeleyev saw that there must exist an element which was unknown at the time. He called it {eka-silicon,} the prefix {eka-} meaning {one} in Sanskrit. By looking at the properties of silicon above and of tin (Sn) below, and also of zinc and arsenic surrounding it, he could guess such properties as its atomic weight, the probable boiling point of some of its compounds, and its specific gravity. In 1886, C. Winkler from the famous mining center of Freiberg in Saxony found the new element in a mineral from the Himmelsfurt mine and called it Germanium. Its actual properties were found to correspond entirely with those forecast by Mendeleyev. There had also been a gap in the Third Group (the 13th in the new system) in the position just under the elements boron and aluminum. In 1871 Mendelyev had named this still unknown element {eka-aluminum.} In 1875, Lecoq de Boisbaudran, using techniques of spectrum analysis, discovered a new metal in a zinc blende ore from the Pyrenees. He named it Gallium. At first it semed to differ considerably from the density Mendeleyev had predicted it would have if it was indeed eka-aluminum. But as observations proceeded the new element was found to possess the density, atomic weight and chemical properties which Mendeleyev had forecast.

That is the essential concept of periodicity. In order for Mendeleyev to arrive at it a great deal of prior chemical investigation was required. Perhaps the most important prerequisite had been the discovery of new elements. The ancients knew 10 of the substances we call elements today, most of them metals. These were iron, copper, lead, tin, antimony, mercury, silver, gold, carbon, and sulfur.[fn 1] All but two of the rest were discovered in the modern era. Between 1735 and 1803, 13 new metals and four gaseous elements were discovered. In 1808 six new elements from the alkali and alkali metal groups (Group 1 and II) were discovered. [fn 2] And the discoveries continued through the 19th century, capped by Marie Curie’s isolation of radium in 1898. In 1869 when Mendeleyev conceived the idea of periodicity about two thirds of the 92 naturally occurring elements were known. Still a few more remained to be discovered in the 20th century. And then came the synthesis of the artificial elements beyond the 92 naturally occurring ones, beginning with neptunium and plutonium.

What do we mean by an element? Chemistry deals primarily with homogeneous substances, not differing in their parts. But the fact that a substance is the same in all its parts does not distinguish it as an element. Sulfur which we consider an element is a yellow powder or cake, but many compounds such as chromium salts can take on a similar appearance. Table salt is uniform and crystalline, but not an element. We consider hydrogen gas an element but carbon dioxide gas a compound. Sometimes elements are described as the elementary building blocks from which more complex substances are formed. But a better definition is the one Lavoisier gave, which describes an element as the result of an action, as that which cannot be further separated by chemical procedures:

“[I]f by the term {elements} we mean to express those simple and indivisible atoms of which matter is composed, it is extremely probable we know nothing at all about them; but, if we apply the term {elements,} or {principles of bodies,} to express our idea of the last point which analysis is capable of reaching, we must admit, as elements, all the substances into which we are capable, by any means to reduce bodies by decomposition. Not that we are entitled to affirm that these substances we consider as simple may not be compounded of two, or even of a greater number of principles; but, since these principles cannot be separated, or rather since we have not hitherto discovered th means of separating them, they act with regard to us as simple substances, and we ought never to suppose them compounded until experiment and observation has proved them to be so.” [fn 3] Lavoisier’s warning remains applicable today. By heeding it, we do not fall into the trap of supposing we are dealing with irreducible elementarities, for the history of scientific progress has shown that increasing mastery over nature always permits us to delve deeper into the microcosm. For chemical technology, the element was the irreducible substance. But later developments allowed us to reach down to the electron, the nucleus, and to subnuclear particles.

It was necessary to perform chemical operations on substances to know if they were elements or compounds. Many things that were once considered elementary were later found to be composite. Lavoisier’s study of the separation of water into hydrogen and oxygen gas, and their reconstitution as water is exemplary. Similarly, his demonstration that the atmospheric air consists primarily of oxygen and nitrogen gas. The metals that were discovered in the 18th century were mostly separated from their ores by processes of chemical reaction, distillation, and physical separation.

At the time Mendeleyev was writing his textbook experimenters had accumulated an enormous store of information concerning the properties of elements and their compounds. Especially of note were the many analogous properties among the elements and their respective compounds. For example, lithium and barium behaved in some respects to sodium and postassium, but in other respects to magnesium and calcium. Looking at such analogies as markers of an underlying ordering principle, Mendeleyev suspected that there must be a way to find quantitative, measurable properties by which to compare the elements. There were four different types of measurable properties of the elements and their compounds, which he took into consideration in formulating his concept of periodicity. He identifies these in Chapter 15 of his textbook as:

(a) isomorphism, or the analogy of crystalline forms; (b) the relations between the “atomic” volumes of analogous compounds of the elements; (c) the composition of their saline compounds; (d) the relations of the atomic weights of the elements.

Think of each of these types of properties as different means of “seeing” into the microcosm. Let us begin with the first, crystal isomorphism. When a compound is dissolved in water or some other solvent, and the water removed by evaporation or other means, it can usually be made to crystallize. All of the familiar gemstones and many rocks are crystals that have been formed under conditions present within or at the surface of the Earth. Table salt and sugar are familiar crystals. Most metals and alloys cool and harden in characteristic crystalline forms. Organic compounds, even living things like proteins, can be made to crystallize for purpose of analyzing their structure. With the development of chemistry following Lavoisier, the crystalline form began to receive more attention, and close study eventually showed that every compound crystallizes in a unique form. Many of these forms are quite similar, but careful measurement of the the facial angles and the proportional lengths of their principal axes will always show some slight difference. Crystallography thus became a means of chemical analysis, and by the 1890s there existed catalogues of the crystallographic properties of nearly 100,000 compounds. [fn 4]

Despite these very fine differences, the general forms of crystals fit into certain classifiable groups. Their shapes include the cube and octahedron, hexagonal and other prisms, and a great number of variations on the Archimedian solids, their duals, and many unusual combination forms. The German chemist Eilhard Mitscherlich first demonstrated in 1819 that many compounds which have similar chemical properties and the same number of atoms in their molecules also show a resemblance of crystalline forms. He called such substances isomorphous. He found that the salts formed from arsenic acid, (H3AsO4) and phosphoric acid (H3PO4), exhibited a close resemblance in their crystalline forms. When the two salts were mixed in solution, they could form crystals containing a mixture of the two compounds. Mitscherlich thus described the elements arsenic and phsophorous as isomorphous.

Following Mitscherlich a great number of other elements exhibiting crystal isomorphism were found. For example, the sulphates of potassium, rubidium and cesium (KSO4, RbSO4, CsSO4) were found to be isomorphic; the nitrates of the same elements were also isomorphic with each other. The compounds of the alkali metals (lithium, sodium, postassium, rubidium) with the halogens (fluorine, chlorine, bromine and iodine) all formed crystals which belonged to the cubic system, appearing as cubes or octahedra. The cubic form of sodium chloride (table salt) crystals is an example, as one can verify with a magnifying glass.

This was the first of the clues which suggested the concept of periodicity. When Mendeleyev arranged the elements in order of increasing atomic weights, the isomorphic substances were found to form one above the next in a single column. Thus arsenic and phosphorous were part of Group V (15, in the modern nomenclature). The alkali metals fell under Group I; the halogens became Group VII (17 in the modern nomenclature). Not only this, but the elements of the same groups combined with one another in the same proportions. Thanks to the work of Gerhardt and Cannizzaro in establishing a uniform system of atomic weights, it had become a simple matter to determine the chemical formula for a great number of substances, once the proportion by weight of the component elements had been determined. It thus turned out that the elements of the first group (designated R) combined with the elements of the seventh group (designated X) in the proportion RX, as in NaCl. The elements of the second group combined with those of the seventh group in the proportion RX2, as in CaCl2, and so forth. If the combinations with oxygen were considered (the oxides being very prevalent), the first group produced RO2, the second group RO, the thrid group R2O3, and so forth. This is what Mendeleyev is describing in the periodic chart we show in Figure 2. We shall save the fascinating question of the investigation of the atomic volumes and many other properties of the elements which prove to be periodic for another time, and end this exercise for now.

NOTES

1. Mining and metallurgy was clearly a part of ancient science, though the thinking and discovery process is mostly lost to us. Heinrich Schliemann, the discoverer of Troy, suggests that the word “metal” came from Greek roots (met’ alla) meaning to search for things, or research. Archaeological remains indicate an ordering of discovery of the metals and the ability to work them, with copper and its alloys preceding iron for example. Ironworking is associated with the Hittite and Etruscan seafaring cultures of Anatolia and north central Italy, who spoke a common language related to Punic or Phoenician.

2. The four gaseous elements were hydrogen (Henry Cavendish, 1766); nitrogen (Daniel Rutherford, 1772); oxygen (Carl Scheele, Joseph Priestley, 1772); chlorine (Scheeele, 1774). Among the metals discovered in the 18th century were:

Platinum (Antonio de Ulloa, 1735); Cobalt (Georg Brandt, 1735); Zinc (Andreas Marggraf, 1746); Nickel (Axel Cronstedt, 1751); Bismuth (Geoffroy, 1753) Molybdenum (Carl Scheele, 1778); Zirconium (Martin Klaproth, 1778); Tellurium (Muller, 1782); Tungsten (Juan and Fausto d’Elhuyar, 1788); Uranium (Klaproth, 1789), Titanium (William Gregor, 1791), Chromium (Louis Vauquelin, 1797); Beryllium (Vauquelin, 1798)

In 1803, William Hyde Wollaston and Smithson Tennant found the elements rhodium, palladium, osmium and irridium in platinum ore. In 1808, Humphry Davy isolated the alkali elements sodium, potassium, magnesium, calcium, strontium, and barium by electrolysis of their molten salts.

3. Antoine Laurent Lavoisier, {Elements of Chemistry,} translated by Robert Kerr, in {Great Books of the Western World,} (Chicago: Encyclopedia Briitannica, 1952) p. 3.

4. In the history of physical chemistry, the study of crystals provided one of the first means of access to the microcosm. It continues to be of importance today.This is great fun because Kepler’s playful work {The Six-cornered Snowflake,} is actually the founding document of modern crystallography. The student must take advantage of this, for the topic, as presented in the usual textbooks, is a confusion of mathematical formalisms and systems of classification. In Kepler, we see that the question is really very simple: why is the snowflake six-sided? why is the beehive made from cutoff rhombic dodecahedra? How shall we get an answer? It can only be by attempting to shape our imagination in conformity with the mind of the creator. If we do not get the complete answer, we see, nonetheless, that it is through the playful exercise of the mind in advancing and pursuing hypothesis that we come closer to it.

Among the many discoveries presented in that small work, Kepler introduces the concept that the study of the close-packing of spheres, which copy the space-filling property of rhombic dodecahedra, can help to explain the mineral crystals, all of which exhibit the characteristic hexagonal symmetries. Kepler thus suggested the existence of an atomic or molecular structure within the abiotic domain. Kepler’s insights were carried forward in the study of mineral crystals especially by the work of the Abbe Hauy (1743-1822) in France, who was followed by a great number of other investigators.

AVOGADRO’S HYPOTHESIS AND ATOMIC WEIGHT (WHEN 2 + 1 = 2)

by Larry Hecht March 29, 2005

(This is the first in a series of pedagogicals which will address the scientific basis of nuclear power from a conceptual, historical standpoint. The figures can be accessed at www.wlym.com/~bruce/atomicweight.zip).

Our modern understanding of the atom and the microscopic domain has its origin in two parallel lines of development in experimental science which date to the period from approximately 1785-1869. This experimental work closely overlaps the developments in mathematics which we have been studying respecting the Gauss-Riemann complex domain, and a patient and not-too-literal approach to its study will lead to many beautiful realizations of the conceptual connections. One track is the Ampere-Gauss-Weber electrodynamics upon which the greater part of modern experimental physics practice rests. The other is the development of the science of chemistry, from the work of Antoine Laurent Lavoisier (1743-1794) to Dmitri Mendeleeff’s 1869 formulation of the periodicity of the elements as arranged by atomic weights. Modern physical chemistry, including nuclear chemistry and the Pasteur-Vernadsky tradition of modern biogeochemistry, owes its existence to these latter developments.

We shall focus here on the second of these two important lines of development.

Most people have heard the term “atomic weight.” What does it mean? To believe in such a notion, we must first accept the existence of a very small, invisible thing called an atom; we must further suppose the existence of common species of atoms, and that each exemplar will exhibit the same properties as any other; finally, we must imagine that we might find some means of weighing this almost non-existent entity. Not only has this proven possible, but, strange as it might seem, the concept of atomic weight lies at the foundation of nearly all the breakthroughs of modern science and technology. Dmitri Mendeleeff’s discovery of the Periodicity of the Elements rests upon this principle, as do all the developments which have allowed us to harness power from the atomic nucleus. The curious anomaly in the atomic weight of helium–that it is less than the sum of the weight of its constituent particles–was the basis for the recognition that fusion energy would be possible. (The possibility of realizing energy from this anomaly, which became known as the “mass defect,” was described in a 1914 paper by William Draper Harkins, the teacher of our friend Dr. Robert Moon). The nuclear chemistry which is the basis for heavy-element fission, the source of power for nuclear reactors, also rests on the concept of atomic weight. Thus, given its importance to the progress of all modern science, we have decided to devote this exercise to an outline of how the concept of atomic weight came about, anticipating that readers will find a way to pursue the further study and experimentation required for a deeper understanding.

A doctrine of atomism–that everything is made up of tiny and indivisible particles–existed since ancient times, its most famous proponent being the Eleatic philospher, Democritus. But the atom of chemistry was not conceived by the followers of Democritus nor his modern reviver Gassendi, but rather by thinkers who tended toward the tradition of Plato, Cusa and Leibniz. The chemical atomic theory, upon which the concept of atomic weight is based, developed in the first decade of the 19th Century out of work centered around the Ecole Polytechnique in Paris. The clear development of the concept of atomic weight took place over the course of several decades following that, led by the inspirer of Mendeleeff, Charles Frederic Gerhardt (1826-1845). Its acceptance was not achieved until a famous international congress of chemists in Karlsruhe in 1860, where an intellectual batttle led by the Italian Stanislao Cannizzarro (1826-1910) finally settled the question.

The experimental development of the concept of atomic weight begins with the study of gases. By the first decade of the 19th century chemists had produced and identified a variety of common gases, including hydrogen, oxygen, nitrogen, chlorine, ammonia, hydrogen chloride, among others. Through techniques pioneered by Lavoisier and refined by subsequent investigators, it was possible to measure quite precisely the volume and weight of gases. When water was decomposed, it could easily be shown that two gases were produced (named by Lavoisier hydrogen and oxygen), in the proportion of two volumes to one (Figure 1–Mendeleeff, Fig. 19, p. 114). By decomposing ammonia by the action of an electric spark, nitrogen and hydrogen gas was produced in the proportion of three volumes to one. Through a great number of experiments with different gases, Joseph-Louis Gay-Lussac (1778-1850) came to the recognition known as his First Law: That the amounts of substances entering into chemical reaction occupy under similar physical conditions in a gaseous or vaporous state, equal or simple multiple volumes. This is also known as the law of combining volumes or the law of multiple proportions. It is a curious result, and hardly an obvious one, if you think about it.

It was also possible to weigh the gases,<fn. 1> and as hydrogen was by far the lightest, it was convenient to compare the weight of a gas to an equal volume of hydrogen. This ratio became known as the vapor density. For oxygen it was 16, for nitrogen 14, for ammonia 8.5, and for water vapour 9. These numbers, too, contained a paradox, for why should a quantity of ammonia, which contains nitrogen, and a quantity of water which contains oxygen, weigh less in proportion to hydrogen than an equivalent volume of the gases they contain? To answer the question, one must have a hypothesis about what a gas is.

Daniel Bernoulli, the son of Leibniz’s collaborator Johann, had proposed an idea in his 1738 book {Hydrodynamics,} which synthesized the research on atmospheric pressure and the pressure and volume relationships of gases that had been carried out by predecesors including Pascal, Torricelli, Mariotte, and Boyle. Bernoulli supposed that a gas, or elastic fluid as he called it, consisted of a great number of tiny, invisible particles which became agitated upon heating, and produced pressure by striking against the walls of its container. Bernoulli’s kinetic theory of gases became enormously important for physical chemistry in the 19th century, even though it was formalized by Clausius, Maxwell, and others into a doctrine (entropy) which was the very opposite of the thinking of its originator. <fn. 2>

Another paradox about the weight and volume of gases arises when we consider the composition of water. We noted that it can be easily shown that in the decomposition of water, two volumes of hydrogen are produced for every volume of oxygen. These gases may be brought back together into a mixture known as detonating gas. When this mixture of two parts hydrogen and one part oxygen is ignited by a spark, an explosion occurs and the product is water. If that quantity of water be vaporized and brought back to the same temperature and pressure of the original gaseous ingredients, it is found that the two volumes of hydrogen plus one volume of oxygen have become two volumes of water vapor! Apparently, 2 + 1 = 2 in the world of water.

[In <fn. 3> I supply Mendeleeff’s description of the apparatus for carrying out this experiment. It is quite detailed, so continue reading, and return to it after you have finished.]

Whether this is paradoxical or not depends not only upon what we think is in the volume of gas, but how much. If we accept Bernoulli’s hypothesis that a gas consists of a great number of tiny, invisible particles, we have still not said how many they are. The first quantitative formulation on this account was proposed by the Italian chemist Count Amedeo Avogadro in a paper published in 1811. Looking at Gay-Lussac’s law and data from his own chemical researches, Avogadro hypothesized that at the same temperature and pressure, equal volumes of any two gases would contain the same number of particles, or molecules as they were coming to be called. His idea was not received with much interest. The only figure of note to embrace Avogadro’s idea, at first, was Andre-Marie Ampere, who was not very well known at the time. <fn. 4> In the 1840s, the French chemist Charles Frederic Gerhardt adopted Avogadro’s hypothesis and labored unceasingly to disprove all doubts concerning its truth. It finally won acceptance at the Karlsruhe Congress in 1860. By 1865, the key to finding the number of atoms or molecules contained in a cubic centimeter of any gas was determined by Josef Loschmidt, then an Austrian high school teacher. <fn. 5>

Employing Avogadro’s hypothesis, we can resolve the paradox of the composition of water, and discover another strange feature of the universe. If the composition of water, as suggested by the volumes obtained upon decomposition, be H20, then one volume of oxygen gas should unite with two volumes of hydrogen gas to produce one volume of the combined gas–H2O. Yet experiment shows the quantity of water vapor produced to be equal to two volumes. The paradox could only be resolved by assuming that the particles of both the hydrogen and the oxygen gas were twins, each consisting of two particles of hydrogen or oxygen. Thus, instead of H and O, the constitutent parts of the two gases must be H2 and O2. Avogadro called them compound molecules; today they are called diatomic molecules. Avogadro found that hydrogen, oxygen, nitrogen and chlorine were of this type and suspected there might be more. Upon detonation, the O2 molecules must break asunder each of the two pieces combining with two hydrogen atoms. Then the description of the formation of two volumes of water from a mixture of detonating gas containing two volumes of hydrogen and one volume of oxygen becomes: 2 H2 + O2 –> 2 H2O.

Avogadro’s conception of a diatomic molecule also served to unravel the paradox noted above concerning the vapor densities. The standard of reference for the vapour density was hydrogen gas, which we have come to see consists of diatomic molecules. Thus, a given volume of hydrogen gas will weigh twice what we would have expected, assuming the constitutents to be single atoms. Oxygen gas, as we have seen from the case of water, is also a diatomic molecule. Thus, the ratio of the weights of equal volumes of oxygen to hydrogen (the vapor density) will correspond to the true ratio of the weights of the atoms, if we assume with Avogadro that equal volumes of gases contain an equal number of constitutent molecules. We can see now, how important it was to establish Avogadro’s hypothesis. Once established, it allows us to infer the relative weights of tiny invisible atoms from the measured weights of large volumes of gases. If we assign the weight of 1 to an atom of hydrogen, we now know that an atom of oxygen will weigh 16. The weight of a molecule of water (H2O) is then 18. But a volume of water vapor weighs only 9 times as much as an equal volume of hydrogen gas, because the hydrogen is diatomic. Nitrogen, it turns out, is also a diatomic gas. Its vapor density of 14 thus denotes its true atomic weight. Ammonia (NH3) has a vapor density of 8.5 and not 17 because it is being compared to diatomic hydrogen gas.

Using Avogadro’s hypothesis, which was rigorously established by Gerhardt in the 1840s, it became possible to establish the atomic weights of a great number of substances, both naturally occurring gases and those substances which could be vaporized.<fn. 6> The unification of the practicing chemists of Europe around the Avogadro hypothesis, which was achieved by Cannizzaro at Karlsruhe in 1860, meant that all the data related to atomic weight could be systematized under one conception, and therefore under one system of measurement. One of the happy results of this achievement was the Periodic Table of the Elements devised by Mendeleeff as an investigation of the peculiar properties of the atomic weights, a topic we shall take up in a future treatment.

– Suggestions on Further Reading: –

I have found the best success in approach to these topics by beginning with a reading of Lavoisier’s {Principles of Chemistry} (available in a Dover paperback edition and as Vol. 45 of the Britannica {Great Books}). In a small weekly telephone meeting, begun about a year and-a-half ago with some interested youth, we completed the Lavoisier text in about 3 to 4 months; some independent experimentation was also carried out during the time. We followed that with a reading of Mendeleeff’s much longer textbook (cited below–parts being scanned in LA for greater access). This reading project is still ongoing. After an initial attempt to jump ahead to Chapter XV, which presents his discovery of the Periodic Table, we returned to page one, taking Mendeleeff’s own advice that a proper appreciation of his discovery requires a grounding in the descriptive and historical aspects of the subject.

NOTES:

(With apologies to Rachel Douglas, I employ the old-style transliteration (Mendeleeff) for consistency with the bibliographic references).

1. Mendeleeff writes in the introduction to his {Principles of Chemistry}: “Gases, like all other substances, may be weighed, but, owing to their extreme lightness and the difficulty of dealing with them in large masses, they can only be weighed on very sensitive balances; that is, on such as, with a considerable load, indicate a very small change in the weight–for example, a milligram in a load of 1,000 grams. In order to weigh a gas, a glass globe furnished with a tight-fitting stop-cock is first of all exhausted of air by an air-pump (a Sprengel pump is the best), after which the stop-cock is closed, and the exhausted globe weighed. If the gas to be weighed is then let into the globe, its weight can be determined from the increase in the weight of the globe. It is necessary, however, that the temperature and presure of the air about the balance should remain the same for both weighings, as the weight of the globe in air varies (according to the laws of hydrostatics) with the density of the latter. The volume of the air displaced, and its weight, must therefore be determined by observing the temperature, density, and moisture of the atmosphere during the time of the experiment. This will be partly explained later, but may be studied more in detail by physics. Owing to the complexity of all these operations, the mass of a gas is usually determined from its volume and its density, i.e. the weight of unit volume.” [D. Mendeleeff, {The Principles of Chemistry,} Third English Edition, translated from the Russian (Seventh Edition) by George Kamensky (London: Longmans Green, 1905) and (New York: Kraus Reprint, 1969), p. 10, note 17.]

2. D. Mendeleeff, op cit., pp. 346-348. Daniel Bernouli, extract from {Hydrodynamica} in Wm. Francis Magie, {A Sourcebook in Physics,} (Harvard Univ. Press, 1963) pp. 247-251. Taken together with Avogadro’s Law (to be explained shortly), the Bernoulli theory leads to the conclusion that under similar conditions of temperature and pressure, gas particles of different mass would each contain the same {vis viva} –the living force of Leibniz, which is measured as one half the product of mass into the square of velocity. The gaseous separation of isotopes, which is used to enrich uranium, makes use of this extension of Leibniz’s original discovery. Refined uranium, which consists of isotopes of two different weights, U-238 and U-235, is combined with fluorine into the gas uranium hexafluoride (UF6). As fluorine has an atomic weight of approximately 19, the hexafluoride gas must contain particles of two different masses, approximately (238 + (6 x 19)) and (235 + (6 x 19)). As the {vis viva} of the particles will be the same at a given temperature and pressure, the U-235-hexafluoride particles must move slightly faster than those of U-238. By pumping the gas through a membrane, a slightly greater concentration of the faster U-235-hexafluoride particles will pass through, and by repeating the process numerous times, separation is achieved.

3. Mendeleeff provides us a description of the apparatus for showing that 2 + 1 = 2 in the world of water. An understanding of the effect of atmospheric pressure on a column of liquid, as established by Pascal and Torricelli, will be necessary to fully comprehend this and all gas volume experiments. The student should be able to master these elementary concepts through self study. Reproducing Dr. Moon’s favorite experiment with atmospheric pressure, as pictured on page 44 of the Fall 2004 “Robert Moon” issue” of {21st Century Science,} will go a long way toward comprehension. Mendeleeff describes the apparatus as follows: “[T]he volume occupied by water, formed by two volumes of hydrogen and one volume of oxygen, may be determined by the aid of the apparatus shown in fig. 64 (Figure 2–from Mendeleef p. 325). The long glass tube is closed at the top and open at the bottom, which is immersed in a cylinder containing mercury. The closed end is furnished with wires like a eudiometer. The tube is filled with mercury, and then a certain volume of detonating gas is introduced. [The gas displaces the mercury being held up in the tube by the atmospheric pressure–LH.] This gas is obtained from the decomposition of water, and therefore, in every three volumes, contains two volumes of hydrogen and one volume of oxygen. The tube is surrounded by a second and wider glass tube, and the vapour of a substance boiling above 100 degrees–that is, whose boiling-point is higher than that of water–is passed through the annular space between them. Amyl alcohol, whose boiling-point is 132 degrees, may be taken for this purpose. The amyl alcohol is boiled in the vessel to the right hand and its vapour passed between the walls of the two tubes. In the case of amyl alcohol the outer glass tube should be connected with a condenser to prevent the escape into the air of the unpleasant-smelling vapour. [In the apparatus pictured the outer glass tube is not connected with a condenser; thus, the puff at the top of the tube is not steam as unfortunately suggested by the caption–LH.] The detonating gas is thus heated up to a temperature of 132 degrees. When its volume becomes constant it is measured, the height of the column of mercury in the tube above the level of the mercury in the cylinder being noted. Let this volume equal {v}; it will therefore contain 1/3 {v} of oxygen and 2/3 {v} of hydrogen. The current of vapour is then stopped and the gas exploded; water is formed, which condenses into a liquid. The volume occupied by the vapour of the water formed has now to be determined. For this purpose the vapour of the amyl alcohol is again passed between the tubes, and thus the whole of the water formed is converted into vapour at the same temperature as that at which the detonating gas was measured; and the cylinder of mercury being raised until the column of mercury in the tube stands at the same height above the surface of the mercury in the cylinder as it did before the explosion [that is, the atmospheric presure in the tube is now the same as before–LH] it is found that the volume of the water formed is equal to 2/3 {v,} that is, it is equal to the volume of the hydrogen contained in it. Consequently the volumetric composition of water is expressed in the following terms: Two volumes of hydrogen combine with one volume of oxygen to form two volumes of aqueous vapour.” [Mendeleeff, op. cit., pp. 325-326]

4. Amedeo Avogadro, “Essay on a Manner of Determining the Relative Masses of the Elementary Molecules of Bodies, and the Proportions in Which They Enter into These Compounds,” {Journal de Physique} 73, 58-76 (1811) [Alembic Club Reprint No. 4] http://web.lemoyne.edu/~GIUNTA/avogadro.html

Andre-Marie Ampere “Lettre de M. Ampere a M. le comte Berthollet, sur la determination des proportions dans lesquelles les corps se combinent d’apres le nombre et la disposition respective des molecules dont leurs particules integrantes sont composees,” {Annales de Chimie,} Tome 90, (30 April 1814) pp. 43-86; 2 planches. Ampere suggests a new series of tetrahedral-based polyhedra which, he suggests, would be the shapes taken by definite compounds.

5. Avogadro’s Number is the number of atoms or molecules of a gas contained in a volume of 22.4 liters at standard temperature and pressure; this volume is used as a reference because it is the volume of a container of hydrogen gas weighing 2 grams. The number of molecules of any gas fitting into such a container at a standard temperature and pressure was determined to be 6.02 x 10 to the 23rd power. This is 602 sextillion molecules, using the American system for naming large numbers, and quite a few by anybody’s count. In a high vacuum of one billionth of an atmosphere, achievable in a laboratory, there remain more than ten billion molecules. Even the so-called vacuum of space is never empty–only less densely populated than other places.

6. Following is the description by Mendeleeff of a method of determining the vapour density of substances which are liquid or solid at ordinary temperature. The method implies knowledge of the relationship of pressure, volume and temperature of gases. Study of the description and diagrams will help the reader to conceptualize the experimental process in working with gases. A study of Lavoisier’s work will help to make the subject clear: “The method by weight is the most trustworthy and historically important. Dumas’ method is typical. An ordinary spherical glass or porcelain vessel, like those shown respectively in figs. 60 and 61 (Figure 3–Mendeleeff p. 321), is taken, and an excess of the substance to be experimented upon is introduced into it. The vessel is heated to a temperature {t} degrees, higher than the boiling-point of the liquid; this gives a vapour which displaces the air, and fills the spherical space. When the air and vapour cease escaping from the sphere, the latter is fused up or closed by some means; and when cool, the weight of the vapour remaining in the sphere is determined (either by direct weighing of the vessel with the vapour and introducing the necessary corrections for the weight of the air and of the vapour itself, or by determining the weight of the volatilised substance by chemical methods), and the volume of the vapour at {t} and at the barometric pressure {h} are then calculated.” [Mendeleeff, op. cit., p. 323 n.]