Elements and Atoms: Chapter 7
Atoms of Definite Weight: Dalton

The laws of definite and multiple proportions are also associated with Dalton, for they can be explained by his atomic hypothesis. The law of definite proportions or of constant composition had previously been proposed in the work of Jeremias Richter and Joseph-Louis Proust [e.g. Proust 1799]. The law of multiple proportions came to be regarded as an empirical law quite independent of its relation to the atomic hypothesis or perhaps as an empirical law that inspired the atomic hypothesis; however, Roscoe and Harden have shown that in Dalton's mind it was a testable prediction which followed from the atomic hypothesis. [Roscoe & Harden 1896]

The selection reproduced in this chapter includes Dalton's atomic hypothesis as he enunciated it in his New System of Chemical Philosophy. It also includes some of Dalton's ideas on heat and gases, ideas which were later to hinder the acceptance of Avogadro's hypothesis (Chapter 9). It is not, however, the first published statement of Dalton's atomic hypothesis or the first table of his atomic weights. A "Table of the relative weights of the ultimate particles of gaseous and other bodies" appears in Dalton 1805b. Thomas Thomson's account of Dalton's atomism [Thomson 1807] also preceded the publication of New System of Chemical Philosophy.

A New System of Chemical Philosophy

Chap. I. On Heat or Caloric[1]

When all surrounding bodies are of one temperature, then the heat attached to them is in a quiescent state; the absolute quantities of heat in any two bodies in this case are not equal, whether we take the bodies of equal weights or of equal bulks. Each kind of matter has its peculiar affinity for heat, by which it requires a certain portion of the fluid, in order to be in equilibrium with other bodies at a certain temperature. Were the whole quantities of heat in bodies of equal weight or bulk, or even the relative quantities, accurately ascertained, for any temperature, the numbers expressing those quantities would constitute a table of specific heats, analogous to a table of specific gravities, and would be an important acquisition to science.[2] Attempts of this kind have been made with very considerable success.

Whether the specific heats, could they be thus obtained for one temperature, would express the relation at every other temperature, whilst the bodies retained their form, is an enquiry of some moment. From the experiments hitherto made there seems little doubt of its being nearly so; but it is perhaps more correct to deduce the specific heat of bodies from equal bulks than from equal weights. It is very certain that the two methods will not give precisely the same results, because the expansions of different bodies by equal increments of temperature are not the same. But before this subject can well be considered, we should first settle what is intended to be meant by the word temperature.

...[3]

CHAP. II. On the Constitution of Bodies.

The opinions I more particularly allude to, are those of Berthollet on the Laws of chemical affinity; such as that chemical affinity is proportional to the mass[7], and that in all chemical unions, there exist insensible gradations in the proportions of the constituent principles. The inconsistence of these opinions, both with reason and observation, cannot, I think, fail to strike every one who takes a proper view of the phenomena.

Whether the ultimate particles of a body, such as water, are all alike, that is, of the same figure, weight, &c. is a question of some importance. From what is known, we have no reason to apprehend a diversity in the particulars: if it does exist in water, it must equally exist in the elements constituting water, namely, hydrogen and oxygen. Now it is scarcely possible to conceive how the aggregates of dissimilar particles should be so uniformly the same. If some of the particles of water were heavier than others, if a parcel of the liquid on any occasion were constituted principally of these heavier particles, it must be supposed to affect the specific gravity of the mass, a circumstance not known. Similar observations may be made on other substances. Therefore we may conclude that the ultimate particles of all homogeneous bodies are perfectly alike in weight, figure, &c.[8] In other words, every particle of water is like every other particle of water; every particle of hydrogen is like every other particle of hydrogen, &c.[9]

Besides the force of attraction, which, in one character or another, belongs universally to ponderable bodies, we find another force that is likewise universal, or acts upon all matter which comes under our cognisance, namely, a force of repulsion. This is now generally, and I think properly, ascribed to the agency of heat. An atmosphere of this subtile fluid constantly surrounds the atoms of all bodies, and prevents them from being drawn into actual contact. This appears to be satisfactorily proved by the observation, that the bulk of a body may be diminished by abstracting some of its heat:[10] But from what has been stated in the last section, it should seem that enlargement and diminution of bulk depend perhaps more on the arrangement, than on the size of the ultimate particles. Be this as it may, we cannot avoid inferring from the preceding doctrine on heat, and particularly from the section on the natural zero of temperature, that solid bodies, such as ice, contain a large portion, perhaps 4/5 of the heat which the same are found to contain in an elastic state, as steam.

We are now to consider how these two great antagonist powers of attraction and repulsion are adjusted, so as to allow of the three different states of elastic fluids, liquids, and solids. We shall divide the subject into four Sections; namely, first, on the constitution of pure elastic fluids; second, on the constitution of mixed elastic fluids; third, on the constitution of liquids, and fourth, on the constitution of solids.

...[11]

Chap. III. On Chemical Synthesis.

Chemical analysis and synthesis go no farther than to the separation of particles one from another, and to their reunion. No new creation or destruction of matter is within the reach of chemical agency. We might as well attempt to introduce a new planet into the solar system, or to annihilate one already in existence, as to create or destroy a particle of hydrogen. All the changes we can produce, consist in separating particles that are in a state of cohesion or combination, and joining those that were previously at a distance.[13]

In all chemical investigations, it has justly been considered an important object to ascertain the relative weights of the simples which constitute a compound.[14] But unfortunately the enquiry has terminated here; whereas from the relative weights in the mass, the relative weights of the ultimate particles or atoms of the bodies might have been inferred[15], from which their number and weight in various other compounds would appear, in order to assist and to guide future investigations, and to correct their results. Now it is one great object of this work, to shew the importance and advantage of ascertaining the relative weights of the ultimate particles,[16] both of simple and compound bodies, the number of simple elementary particles which constitute one compound particle, and the number of less compound particles which enter into the formation of one more compound particle.[17]

If there are two bodies, A and B, which are disposed to combine, the following is the order in which the combinations may take place, beginning with the most simple: namely,

1 atom of A + 1 atom of B = 1 atom of C, binary.

1 atom of A + 2 atoms of B = 1 atom of D, ternary.

2 atoms of A + 1 atom of B = 1 atom of E, ternary.

1 atom of A + 3 atoms of B = 1 atom of F, quarternary.

3 atoms of A + 1 atom of B = 1 atom of G, quarternary.

&c. &c.

The following general rules may be adopted as guides in all our investigations respecting chemical synthesis.

1st. When only one combination of two bodies can be obtained, it must be presumed to be a binary one, unless some other cause appear to the contrary.

2d. When two combinations are observed, they must be presumed to be a binary and a ternary.

3d. When three combinations are observed, they must be presumed to be a binary, and the other two ternary.

4th. When four combinations are observed, we should expect one binary, two ternary, and one quarternary, &c.[18]

5th. A binary compound should always be specifically heavier than the mere mixture of its two ingredients.[19]

6th. A ternary compound should be specifically heavier than the mixture of a binary and a simple, which would, if combined, constitute it; &c.

7th. The above rules and observations equally apply, when two bodies, such as C and D, D and E, &c. are combined.

From the application of these rules, to the chemical facts already well ascertained, we deduce the following conclusions; 1st. That water is a binary compound of hydrogen and oxygen, and the relative weights of the two elementary atoms are as 1:7, nearly; 2d. That ammonia is a binary compound of hydrogen and azote, and the relative weights of the two atoms are as 1:5, nearly; 3d. That nitrous gas is a binary compound of azote and oxygen, the atoms of which weigh 5 and 7 respectively;[20] that nitric acid is a binary or ternary compound according as it is derived, and consists of one atom of azote and two of oxygen, together weighing 19; that nitrous oxide is a compound similar to nitric acid, and consists of one atom of oxygen and two of azote, weighing 17;[21] that nitrous acid is a binary compound of nitric acid and nitrous gas, weighing 31; that oxynitric acid is a binary compound of nitric acid with oxygen, weighing 26; 4th. That carbonic oxide is a binary compound, consisting of one atom of charcoal, and one of oxygen, together weighing nearly 12; that carbonic acid is a ternary compound, (but sometimes binary) consisting of one atom of charcoal, and two of oxygen, weighing 19;[22] &c. &c. In all these cases the weights are expressed in atoms of hydrogen, each of which is denoted by unity.

In the sequel, the facts and experiments from which these conclusions are derived, will be detailed; as well as a great variety of others from which are inferred the constitution and weight of the ultimate particles of the principal acids, the alkalis, the earths, the metals, the metallic oxides and sulphurets, the long train of neutral salts, and in short, all the chemical compounds which have hitherto obtained a tolerably good analysis. Several of the conclusions will be supported by original experiments.

From the novelty as well as importance of the ideas suggested in this chapter, it is deemed expedient to give plates, exhibiting the mode of combination in some of the more simple cases. A specimen of these accompanies this first part. The elements or atoms of such bodies as are conceived at present to be simple, are denoted by a small circle, with some distinctive mark;[23] and the combinations consist in the juxta-position of two or more of these; when three or more particles of elastic fluids are combined together in one, it is supposed that the particles of the same kind repel each other, and therefore take their stations accordingly.[24]

END OF PART THE FIRST.

TABLE OF ATOMIC WEIGHTS[25]

Notes

[2]Dalton distinguishes between heat and temperature--a distinction which is valid even though Dalton was mistaken about the nature of heat. The amount of heat required to raise the temperature of a gram of water (from 20°C to 21°, for example) is different from the amount of heat required to move the temperature of a gram of aluminum through the same temperature interval. We say that these materials differ in heat capacity or specific heat. Dalton is proposing here a program of systematically tabulating heat capacities. By the way, bulk here means volume, so Dalton ponders whether a comparison of equal masses or equal volumes of material is more appropriate.

[3]I have omitted the sections of chapter I that follow these introductory paragraphs. --CJG

[4]"Bodies of sensible magnitude" means macroscopic objects, that is objects (bodies) large enough to be observed (sensed).

[5]To Dalton, an atom was the smallest piece of matter which retained its identity as a particular substance. He did not observe the distinction modern scientists make between atoms and molecules. Thus, he could speak of an atom of water, meaning the smallest piece of water that was still water; if it was divided further, it would no longer be water, but rather hydrogen and oxygen. Dalton called the smallest particles of chemical compounds "compound atoms" (molecules in modern terminology), and the smallest particles of chemical elements "simple atoms" (now just atoms).

[6]The gaseous state contains the same substance as the solid and liquid states. If one were considering gases without regard to solids or liquids, the picture of a tenuous elastic continuum may be at least as appealing as that of widely spaced particles. The continuum picture is difficult to reconcile with the transformation of a gas to a liquid or solid, though, for the question naturally arises of why the gas is so spread out and the solid or liquid so compact. The atomic picture has a ready-made model for solids and liquids as closely spaced arrangments of the tiny particles and for gases as those same particles spread out through space. Here Dalton simply notes that the atoms in a gas are not held together by the powers of attraction that keep a solid compact. Below and elsewhere, however, he examines why he believes the atoms of a gas repel each other, namely because of their supposed coat of caloric [Nash 1957; see also chapter 5, note 17]. View Dalton's illustration of gases surrounded by a coating of caloric (University of Illinois at Urbana-Champaign).

[7]Dalton passes with no transition from the physical cohesion associated with condensation to the chemical cohesion involved in the formation of compounds. Both phenomena come under the heading of affinity. Dalton refers to Claude Louis Berthollet's ideas on the proportions by mass of elements in compound bodies. Berthollet believed that variable proportions were the rule and definite proportions the exception. Many of Berthollet's variable-proportion "compounds" are actually mixtures, particularly solutions or alloys. (The exceptions, true variable-proportion compounds, are now called berthollides.) At this time, then, the concepts of chemical affinity and chemical compound included phenomena which are outside the scope of the terms' current meanings.

Dalton refers in particular to Berthollet's law of mass action, which retains a place in modern theories of chemical equilibrium, but not in chemical composition. Berthollet recognized that the extent to which a chemical reaction took place depended on the relative amounts of the reactants. He incorrectly believed that the proportions of the reactants would affect the composition of a compound (i.e., the proportions of its components). The effect of the proportions of the reactants really is on the extent of formation of a compound of fixed composition.

[8]Dalton's words recall those of Isaac Newton (1642-1727; view portrait by Godfrey Kneller at the National Portrait Gallery, London) on atoms [Newton 1704]:

"All these things being consider'd, it seems probable to me, that God in the Beginning form'd Matter in solid, massy, hard, impenetrable Particles, of such Sizes and Figures, and with such other Properties, and in such Proportion to Space, as most conduced to the End for which he form'd them; and that these primitive Particles being Solids, are incomparably harder than any porous Bodies compounded of them; even so very hard, as never to wear or break in pieces; no ordinary Power being able to divide what God himself made one in the first Creation. ..."

Indeed, Dalton had copied this Query of Newton's into his notebook with some words to the effect that a working hypothesis can be useful in suggesting lines of inquiry [Knight 1967, p. 31]. Dalton's atoms may be conceived as similar to Newton's, but with different kinds of atoms corresponding to different elements.

[9]What Dalton found "scarcely possible to conceive" is actually true: not all molecules of water are exactly alike, because not all atoms of oxygen are exactly alike nor are all atoms of hydrogen exactly alike. The different forms of atom of the same element are called isotopes. (See chapter 20.) In Soddy's words, "their atoms have identical outsides but different insides." [Soddy 1922]

Dalton dismissed the possibility of isotopes because he reasoned that different samples of water would have observably different properties (weights, for example), if there were different kinds of water molecules. Such a situation was not known to anyone in the early 19^th century, and Dalton believed no such situation existed. There were at the time no processes which either naturally or artificially separated isotopes to a detectable degree, so all substances in which a particular element appeared had the same proportion of isotopes. Thus, for example, every sample of water was perceived to be identical not because every water molecule was identical, but because no water sample had detectably different proportions of the various kinds (or isotopomers) of water molecules.

[10]That is, most materials contract when they are cooled.

[11]I have omitted the sections of chapter II that follow these introductory paragraphs. --CJG

[12]Dalton imagined (correctly) that molecules in a gas are for the most part widely separated from one another.

Dalton's language was a bit more metaphorical than modern technical writing, but anthropomorphic language has not completely disappeared. Dalton's "atoms" supported their dignity by keeping others at a respectful distance; modern molecules are often described as "attacking" reactive sites on "target" molecules.

[13]In this seemingly innocuous paragraph, Dalton combined the principle of conservation of matter with his atomic hypothesis to produce a picture of chemical reactions that is no less than second nature to modern chemists. That is, Dalton took the fact that matter was neither created nor destroyed in chemical processes and related it to the concept of atoms: if atoms (simple atoms, at any rate) were indestructible particles, then no matter would be created or destroyed because no particles of matter would be created or destroyed. All of this, so far, is implicit in Newton's view of atoms (note 8). If simple atoms were indestructible, then perhaps chemical reactions only involved the rearrangements of atoms. Today, this image of chemical reactions joining and separating atoms is so firmly ingrained that it is difficult to imagine any other way of picturing chemical reactions.

[14]In the quantitative study of chemistry, many researchers were engaged in measuring "combining weights." For example, by Dalton's measurements, it appeared that one gram of hydrogen would react with seven grams of oxygen to produce eight grams of water. (Actually, the ratio is much closer to 1 g hydrogen plus 8 g oxygen, yielding 9 g water.) Since oxygen and hydrogen always combined in a ratio of 8:1 by mass, one could say that eight grams of oxygen was equivalent to one gram of hydrogen--at least for the purposes of their reaction to make water.

[15]Dalton proposed compiling and comparing combining weights from different reactions to infer the relative combining weights of all the elements then known. For example, eight grams of oxygen and three grams of carbon react to form eleven grams of a compound (now known as carbon dioxide). Since eight grams of oxygen is equivalent (in one reaction) to three grams of carbon and (in another reaction) to one gram of hydrogen, then Dalton's hypothesis would suggest that three grams of carbon and one gram of hydrogen are equivalent to each other. Thus, if carbon and hydrogen combine with each other, they ought to form a compound with the proportions of three grams carbon to one hydrogen (which they do). So Dalton proposed to construct a table of combining weights of elements and compounds, which he hypothesized were proportional to the weights of their respective simple atoms and compound atoms. Dalton's originality was not in compiling a table of equivalent weighs, but in associating those weights with those of atoms. Note, however, that on a modern periodic table, where the weight of hydrogen is approximately 1, the weight of carbon is 12 (not 3) and that of oxygen is 16 (not 8). Dalton's proposal here was not exactly right, as we shall see, but it was fruitful because it was a step in the right direction. [Giunta 2001]

[16]Dalton's proposal here is to compile a list of atomic weights. In hindsight it is easy to say that this focus on mass rather than shape ("figure") or any other property of atoms was sound because it was fruitful. After all, even those 19^th-century chemists who doubted the literal existence of ultimate particles, found the concept of equivalent weights useful; chemists evolved an ever more powerful set of analytical tools for determining chemical compositions and combining weights. Mass was a sound property to investigate for two reasons: it was accessible experimentally with the technology of the time, and it was a property that was easily quantified and therefore capable of being arranged ordinally.

[17]The determination of chemical formulas would be facilitated by a table of atomic weights. For example, if the equivalent weights of copper and oxygen were 32 and 8 respectively, and if a certain substance required 16 grams of oxygen to combine with 32 of copper, then that substance must be made of two atoms of oxygen for every atom of copper.

[18]This is where modern chemists can see, with hindsight, the flaw in Dalton's scheme. Even contemporary critics of Dalton recognized this scheme to be arbitrary [Avogadro 1811; Knight 1967, p. 18].

In modern terms, Dalton wishes to determine molecular formulas and atomic weights at the same time. If one knows molecular formulas, then it is possible to derive atomic weights from combining weights. But Dalton does not know the formulas of compounds. So he uses the plausible but arbitrary rules described here to guess at formulas. For example, in Dalton's time, three different compounds of nitrogen with oxygen were known. Dalton would have written their formulas as NO, NO₂, and N₂O if he used modern symbols for the elements; indeed, these are the correct formulas. At the same time, only one compound of oxygen and hydrogen was known (water) and only one of nitrogen and hydrogen (ammonia). Dalton's formulas for these substances would be HO and HN; however, the correct formulas are H₂O and NH₃.

[19]By "specifically heavier," Dalton meant "denser" (more weight in a given amount of space, or less space for a given amount of weight). This prediction follows from Dalton's picture of chemical compounds as the intimate union of their contituent atoms. A collection of NO molecules, it stands to reason, must weigh as much as a collection of the same number of N atoms and O atoms, but take up less space because the atoms are held in intimate contact.

[20]"Nitrous gas" is known today as nitric oxide, NO. Note the utility Dalton's program has in principle. From the first two results, one can predict that the ratio of oxygen to nitrogen by weight in nitrous gas is 7:5 or 1.4. The actual ratio, however, is 1.14, considerably lower.

This is a good place to note that if Dalton had had access to accurate and precise analyses, his formulas would not have been self-consistent; his formulas only appeared to be plausible because the analyses contained a large degree of experimental error. The analysis, remember, is the result of an experimental measurement, and does not depend on the atomic hypothesis. Consider modern analyses of water and ammonia, and apply Dalton's hypotheses to them. By weight, the ratio of oxygen to hydrogen in water is 7.94:1; the ratio of nitrogen to hydrogen in ammonia is 4.63:1. Now Dalton decided to use hydrogen as the unit for his system of atomic masses. Since he assumed the formulas of water and ammonia were HO and NH respectively, he would suppose the atomic weights of oxygen and nitrogen were 7.94 and 4.63, respectively, if he had access to modern analyses. Now consider a testable prediction based on Dalton's hypotheses: if the atomic weights of oxygen and nitrogen are the ones just derived, and if the formula for "nitrous gas" is (as Dalton believed it was) NO, then the predicted mass ratio of oxygen to nitrogen in "nitrous gas" should be 1.71:1, far lower than the actual value of 1.14:1. The best analyses to which Dalton had access put the ratio between 1.15:1 and 1.36:1 [Dalton 1810]. The point here is that the analyses were sufficiently imprecise to make the errors embodied in the combination rules less apparent.

[21]"Nitric acid" is known today as nitrogen dioxide, NO₂; "nitrous oxide" is still called nitrous oxide, N₂O. Dalton got these formulas right.

[22]"Charcoal" means carbon. "Carbonic oxide" is known today as carbon monoxide, CO; "carbonic acid" is known today as carbon dioxide, CO₂. Dalton got these formulas right.

[23]These atomic symbols never caught on. As might be imagined, they were too cumbersome for printing or even for writing. The modern symbols of the elements, one- or two-letter abbreviations of elements' names, were introduced early in the 19^th century by the Swedish analytical chemist Jöns Jacob Berzelius. [Berzelius 1813-14]

[24]For example, in the compound "carbonic acid," Dalton displays the two oxygen atoms on opposite sides of the carbon. This happens to be correct, but it is not correct to suppose that in gases particles of the same kind of gaseous element repel each other. For example, in nitrous oxide, the actual structure has two adjacent nitrogen atoms. When Avogadro [Avogadro 1811] later proposed that common gaseous elements existed as diatomic molecules (for example, N₂ or O₂), Dalton and many others could not accept the idea, believing that the like atoms would repel each other in a gas.

[25]This table of atomic weights, not surprisingly, bears little resemblance to current tables. Entries 1-20 are supposed to be of simple atoms, i.e. elements. In fact, items 7-12 are oxides which had not yet been decomposed, as mentioned in chapter 3. Humphry Davy's application of electrical methods of decomposition isolated the metals magnesium, calcium, sodium, potassium, strontium, and barium respectively from these oxides at roughly the same time as Dalton wrote his book [Davy 1808a, 1808b].

Among the compound atoms, there are several unfamiliar names not mentioned in the main text above. Olefiant gas is ethene, C₂H₄. Carburetted hydrogen is methane, CH₄, the chief component in natural gas. Sulphuretted hydrogen is hydrogen sulfide, H₂S, the odor of rotten eggs. Acetous acid is acetic acid, CH₃COOH, the active ingredient in vinager. Finally, Dalton chooses to write some of the more complex compounds as composed of simpler compounds. For example, he presents sugar as a compound of ethanol and carbon dioxide, consistent with the course of fermentation, which breaks sugar down into ethanol and carbon dioxide.

References

• 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, 58-78 (1811); annotated excerpts
• Jöns Jacob Berzelius, "Essay on the Cause of Chemical Proportions, and on Some Circumstances Relating to Them: Together with a Short and Easy Method of Expressing Them," Annals of Philosophy 2, 443-454 (1813), 3, 51-2, 93-106, 244-255, 353-364 (1814)
• John Dalton, "Experimental Enquiry into the Proportion of the Several Gases or Elastic Fluids, Constituting the Atmosphere," Memoirs of the Literary and Philosophical Society of Manchester 1, 244-58 (1805a)
• John Dalton, Memoirs of the Literary and Philosophical Society of Manchester, 1, 271-87 (1805b)
• John Dalton, A New System of Chemical Philosophy Part I (Manchester, 1808); annotated text above.
• John Dalton, A New System of Chemical Philosophy Part II (Manchester, 1810), pp. 316-9.
• Humphry Davy, "The Bakerian Lecture: On Some New Phenomena of Chemical Changes Produced by Electricity, Particularly the Decomposition of the Fixed Alkalies, and the Exhibition of the New Substances Which Constitute Their Bases; And on the General Nature of Alkaline Bodies,"Philosophical Transactions of the Royal Society 98, 1-44 (1808a)
• Humphry Davy, "Electro-Chemical Researches, on the Decomposition of the Earths; With Observations on the Metals Obtained from the Alkaline Earths, and on the Amalgam Procured from Ammonia," Philosophical Transactions of the Royal Society 98, 333-70 (1808b)
• Carmen J. Giunta, "Using History to Teach Scientific Method: the Role of Errors," Journal of Chemical Education 78, 623-627 (2001)
• David M. Knight, Atoms and Elements (London: Hutchinson, 1967)
• Leonard K. Nash, "The Atomic-Molecular Theory" in James Bryant Conant, ed., Harvard Case Histories in Experimental Science, Vol. 1, (Cambridge, MA: Harvard, 1957), pp. 215-321.
• Isaac Newton, Opticks (London, 1704), query 31
• Joseph-Louis Proust, "Researches on Copper," Annales de chimie 32, 26-54 (1799)
• Henry E. Roscoe & Arthur Harden, A New View of the Origin of Dalton's Atomic Theory, (London: Macmillan, 1896)
• Frederick Soddy, Nobel Lecture, December 12, 1922; in Nobel Lectures: Chemistry, Vol. I, 1901-1921 (Amsterdam: Elsevier, 1966), pp. 371-99
• Thomas Thomson, System of Chemistry (Edinburgh, 1807), 3^rd ed., vol. 3, pp. 424-9