Document created: 13 January 04
Air University Review,
MarchApril 1972
Lieutenant Colonel Herman L. Gilster
One of the classic campaigns in the annals of military history was waged at Chancellorsville, Virginia, in May 1863 between the Army of the Potomac, led by Major General Joseph L. Hooker, and the Army of Northern Virginia, commanded by General Robert E. Lee. During the campaign, Lee, with a force approximately half the size of Hooker’s, repulsed the North’s advance into Virginia and achieved a strategic victory that has been studied by students of military art throughout the world. However, today’s critics of the quantitativeoriented decision tools being used by our military services say that this battle would never have transpired if these same tools had been used then.^{1} They feel that under the present decisionmaking process Lee would not have met Hooker’s advance but instead would have retreated to southern Virginia or even into North Carolina. Contrary to that course, Lee decided to give battle, and he won a brilliant victory.
The question to which we must address ourselves, then, is this: Was Lee’s decision to fight based strictly on native intuition—leaving quantitative analysis nothing to offer—or could it be rationally justified by using modern decision techniques? This article argues that there are decision tools inbeing today that can be used to support Lee’s decision. Whether Lee applied such tools, either consciously or subconsciously, is not known, but we do know that he was no stranger to the science of numbers. Douglas Southall Freeman, who spent over twenty years studying the life of the great Confederate commander, declared: “His mind was mathematical and his imagination that of an engineer.”^{2}
Lee’s background supplies ample evidence to confirm this evaluation. He graduated second in the class of 1829 from West Point, which was at that time primarily an engineering school. So proficient had he been in the field of mathematics that he was appointed acting assistant professor to instruct other cadets when he was only a secondyear student. After graduation he entered the Corps of Engineers, and subsequent years found him working on engineering projects throughout the United States. It is doubtful if a person as familiar with numbers as Lee would not either explicitly or implicitly have quantified at least partially the alternatives open to him at Chancellorsville.
In the following sections the reader will find descriptions of three decision tools that could have been applied by Lee to support his decision to fight at Chancellorsville. These tools are the Lanchester equations, Bayes’ theorem, and the von NeumannMorgenstern utility theorem. Before these decision tools are outlined, however, a brief description of the battle of Chancellorsville may prove useful.
Probably the most comprehensive and unbiased study of this battle appears in The West Point Atlas of American Wars, edited by Colonel Vincent J. Esposito, former Professor of Military Art and Engineering at West Point.^{3} The following description draws heavily upon that fine work.
In April 1863 the newly appointed commander of the Army of the Potomac, General Hooker, with 118,000 men, faced General Lee’s Army of Northern Virginia, approximately 60,000 strong, across the Rappahannock River at Fredericksburg, Virginia. On the 29th and 30th Hooker moved approximately 73,000 troops on a wide flanking movement across the Rappahannock to the vicinity of Chancellorsville to attack Lee from the rear. To hold Lee in position, Major General John Sedgwick, U.S. Army, with the remaining 45,000, maintained his position opposite Fredericksburg. (Figure 1a) Although Hooker’s units were in position on the 30th, he awaited further reinforcements and did not advance from the vicinity of Chancellorsville until the first of May.
By this time Lee had interpreted Hooker’s strategy. Leaving Major General Jubal Early, C.S.A., with 10,000 men to face Sedgwick, Lee moved his units toward Chancellorsville. The first clash occurred the afternoon of the first, and Hooker, apparently having lost his courage, gave up the initiative and recalled his much larger force to Chancellorsville into a defensive position.
That night Lee and Lieutenant General “Stonewall” Jackson, aware of Hooker’s hesitancy, conceived a daring plan. Lee would maintain his position with approximately 17,000 men and demonstrate against Hooker’s front, while Jackson would take the remaining force, using Major General Jeb Stuart’s cavalry as a screen, and turn the enemy flank. (Figure 1b)
The movement took the better part of the next day, but shortly before sundown Jackson struck Hooker’s exposed flank. The battle raged during the night until the Federal Army gave way before Jackson’s thrusts. The sensation of victory that Lee felt, however, must have been more than overshadowed by the loss of Jackson, who had ridden too far forward in reconnoitering the Union positions and had been shot by mistake when returning to his own lines.
On the third of May, Hooker again failed to take the initiative against Lee’s split army, and although he was wounded later in the day by cannon fire, he would not relinquish command to his subordinate. By sundown Lee had united his separated units and was pushing Hooker back against the Rappahannock. But Lee’s troubles were not over. Earlier that day Sedgwick had attacked at Fredericksburg, overrun Early’s weak position, and was marching toward Lee’s rear.
Figure 1 The battle of Chancelorville 
Again counting on Hooker’s hesitancy, Lee reversed his field, leaving Jeb Stuart with 25,000 men to face Hooker’s 73,000, and marched the remaining units toward Sedgwick’s advancing army. Another flanking movement, using Early’s remaining force, proved successful, and the morning f the fifth found Sedgwick back across the Rappahannock. (Figure 1c)
Lee, determined to crush Hooker, again reversed his field. But Hooker had had enough. On the sixth of May he withdrew his forces across the river before Lee could accomplish this objective.
A rather mathematical approach to the problem of battle decisions was provided by Frederick Lanchester in his article, “Mathematics in Warfare.”^{4} He derived two basic equations relating numerical strength and another constant, which he called “fighting value,” to total strength. These equations can be adapted to the present analysis if we let “fighting value” represent the aggregate of all factors affecting the battle other than numerical strength.
Lanchester assumed that the number of men killed or incapacitated per unit time during a battle is directly proportional to the strength of the opposing force. This can be shown mathematically as

in which b and r represent the numerical strengths of the Blue and Red forces, respectively; t is time; and K_{b} and K_{r} are the fighting values of the two units.
If K_{b}= K_{r}, the battle depends entirely on the numerical strengths of the two forces. If Blue has twice as many men as Red, the ensuing battle is as depicted in Figure 2a. When Red’s force has been completely annihilated, Blue will have 866 men remaining.

Incidentally, this also shows the value of concentration. If Red originally had 1000 men and separated them into two armies, and each gave battle in turn, Blue would have 866 men after destroying the first Red army—enough easily to defeat Red’s second force, all other things remaining equal.
If the values K_{b} and K_{r} are not equal, then these, too, must be considered in equating the total strengths of the forces. For the condition of equality, losses must be proportional to numerical strengths:
In words, the total strengths of the two forces are equal when the squares of the numerical strengths, multiplied by the fighting values of the units, are equal. This is what Lanchester called the “nsquare Law.”
The effect of concentration versus separation of forces has already been mentioned. Lanchester also gave a mathematical relationship for the aggregate numerical strength of the separated forces. (Figure 2b) Let the numerical values of the Blue and Red forces be represented by lines b and r. In an infinitesimal interval of time the change in b and r will be represented by db and dr in the relationship:
Since in the “nsquare Law” we are interested in the squares of the strengths, we here note what happens to the change of the area of b^{2} and r^{2} when the increments db and dr are subtracted. The change in b^{2} is 2bdb and the change in r2 is 2rdr. According to equation (5) these are equal, so the difference between the two squares is constant.
r. represents numerically a second Red army of the strength necessary in a separate action to place the Red forces on equal terms with the Blue force. Graphically, Red’s total numerical strength is the hypotenuse of a right triangle, the legs of which are the two separate forces. (Figure 2c)
Now if Lee had had available Lanchester’s equations (4) and (6), he could have mathematically verified his decision to fight. First, let us compare the K values. The battle of Fredericksburg, which was the last engagement between the two armies, provides a starting point. The numerical strengths and losses of the Northern forces were both twice that of the South. Accordingly, equation (3) is satisfied, and there existed an equality in the total fighting strength of both sides. By equation (4),
Lee had a fourtoone advantage in “fighting value.”
At the time of Lee’s critical decision, Hooker had divided his army into two forces. One force of approximately 45,000 men under Sedgwick was left to contain Lee, while Hooker, with 73,000 men, effected a flanking maneuver to attack Lee’s rear. In the meantime, however, Lee had left 10,000 men in place under Early to face Sedgwick and took 50,000 men to meet Hooker’s main thrust. According to equation (6), the proportional numerical strengths were then:
Lee’s total strength was greater than Hooker’s!
I hesitate to push this approach too far. The K values were derived from only one campaign and would require further verification. The analysis is predicated on the assumptions that the separated forces give battle in turn and that combat takes place in the open. The first assumption was fulfilled at Chancellorsville, but the second might prove difficult to verify. The approach does show, however, that the Lanchester equations, even if indiscriminately applied, could be used to support Lee’s decision.
Bayes’ theorem can be utilized to refine any hypothesis that Lee might have held about defeating the Northern forces. One version of this theorem takes the form:

If Lee had placed a certain a priori probability on the hypothesis that he could defeat the Northern army, and if the probability of winning a battle, given that the hypothesis was true, was relatively high whereas the normal probability of winning was relatively low, then given the past event—the battle of Fredericksburg (or better yet, eleven wins in thirteen encounters)—his a posteriori probability of the hypothesis would be greater and more meaningful than his a priori probability.
For example, let us say Lee placed a .3 probability on the hypothesis that he could defeat the Union force. If the probability of winning at Fredericksburg, given the hypothesis was true, was .6, and a normal probability of winning at Fredericksburg was .2, then
His a priori probability of winning was .3, but with the use of additional information (past events), this probability increased to .9. He would now have greater faith in his original hypothesis that he could defeat the Union army and might therefore decide to meet Hooker’s advance.
Professors John von Neumann and Oskar Morgenstern have shown that under certain circumstances it is possible to construct a set of numbers for a particular individual that can be used to predict his choice in uncertain conditions. Briefly, this theorem states that if an individual can rank three commodities in an order of preference, say A>B>C, then in a choice between a certain prospect containing B and an uncertain prospect containing A and C with a probability, p, of getting A, there is a value p which makes the individual indifferent between the two prospects. Two of the commodities can be given arbitrary values, and once the individual provides the probability, p, which makes him indifferent between the two prospects, the value of the third commodity can be obtained. These values will then have certain cardinal properties that can be used to evaluate the decision process.
Napoleon stated that “the General is the head, the whole of the Army.”^{5} If Napoleon’s maxim is correct, Lee’s decision to fight could have been predicated on a comparison of the highlevel commanders of the two armies. He did know a majority of the commanders on both sides. Of the eight corps commanders under Hooker, five had served with Lee in the Mexican War and two had been cadets at West Point when Lee served as superintendent of that institution from 1852 to 1855. Aligned against these commanders, Lee had the following men who would playa significant role in the coming battle: Jackson, the trusty lieutenant who had more than proved himself in previous campaigns; Stuart, the dashing cavalry officer who had highly impressed Lee as a cadet at West Point; and Early, an 1837 classmate of Hooker and a veteran of the Mexican War.
That Lee had definite opinions about the abilities of his enemy is apparent from the letters of that day. Previously, when McClellan had been replaced as commander of the Army of the Potomac, Lee expressed sorrow that his old associate of the Mexican War would no longer oppose him: “We had always understood each other so well. I fear they may continue to make these changes till they find someone whom I don’t understand.”^{6} When Hooker replaced Burnside as commander of the federal forces, Lee accepted the change with complacency. In his personal letters, however, he jested mildly over the apparent inability of Hooker to determine a course of action.^{7}
Contrasted with this rather low opinion of the opposition leader, we find this lofty estimate of Jackson’s capabilities: “Such an executive officer the sun never shone on. I have but to show him my design, and I know that if it can be done, it will be done. No need for me to send or watch him. Straight as a needle to the pole he advances to the execution of my purpose.”^{8}
This intimate knowledge of the opposing commanders and definite opinion of their capabilities belong in Lee’s calculus. Given this, he could have used the von NeumannMorgenstern utility theorem to establish a quantitative comparison of the leadership abilities of both sides. As an example, let us say that Lee would rate the three commanders, Hooker, Jackson, and Stuart, in the following order: Jackson>Stuart>Hooker. We now set any arbitrary value for Jackson, say 100, and Stuart, say 90, and then determine at what probability, p, Lee would be indifferent between the certain prospect of getting Stuart and the uncertain prospect which, if selected, provided the probability, p, of getting Jackson.
Numerical values of the capabilities of the other commanders could be derived in the same manner. These values could then be aggregated to give a rough quantitative comparison of Lee’s view of the opposing leadership abilities. This comparison would provide an important input to the decisionmaking process.
This article describes how three modern quantitative tools could have been employed by General Robert E. Lee to aid in the critical decision facing him on the eve of the battle of Chancellorsville. This survey of decision tools is certainly not exhaustive—there are others that one could utilize. There are also other inputs that belong in Lee’s calculus, such as the “super” image of Lee that had been created, the effect of the recently issued Emancipation Proclamation in hardening Southern resistance, and the comparative morale in the two armies.
The point to be emphasized, however, is that any tool, quantitative or otherwise, which aids the decisionmaker in his choice, not only should but must be employed. If that choice is among a number of alternatives, however, systematic quantitative analysis will prove essential in delineating clearly the basic relationships and interactions between the many diverse factors that the decisionmaker must consider. It will prove even more essential in the military than in the business world, where the forces of competition working through the price mechanism furnish a reliable guide to planning. In matters of national security no such mechanism is available.
This does not mean that sound judgment has been replaced by the computer. As far as I can determine, no one has ever advocated the exclusive use of mathematical tools in the determination of policy. Surely this was not the theme of the Hitch and McKean book, which had such a powerful impact upon defense strategy:
Economic choice is a way of looking at problems and does not necessarily depend upon the use of any analytical aids or computational devices. . . . Where mathematical models and computations are useful, they are in no sense alternatives to or rivals of good intuitive judgment; they supplement and complement it. Judgment is always of critical importance in designing the analysis, choosing the alternatives to be compared, and selecting the criterion. Except where there is a completely satisfactory onedimensional measurable objective (a rare circumstance), judgment must supplement the quantitative analysis before a choice can be recommended.^{9}
The responsibility of decision still rests with the commander. Quantitative analysis does not relieve him of that responsibility, but it can make that responsibility less formidable. Hq Pacific Air Forces
Notes
1. One such view was expressed by Colonel Francis X. Kane in “Security Is Too Important to Be Left to Computers,” Fortune, April 1964.
2. Douglas S. Freeman, Lee, abridgement by Richard Harwell (New York: Charles Scribner’s Sons, 1961), p. 115.
3. Vincent J. Esposito, ed., The West Point Atlas of American Wars, Vol. I (New York: Frederick A. Praeger, 1959), pp. 8497.
4. Frederick W. Lanchester, “Mathematics in Warfare,” The World of Mathematics, Vol. IV, edited by James R. Newman (New York: Simon and Schuster, 1956), pp. 21446.
5. Esposito, p. 91.
6. Freeman, p. 268.
7. Ibid., p. 285.
8. Ibid., p. 292.
9. Charles J. Hitch and Roland N. McKean, The Economics of Defense in the Nuclear Age (Cambridge: Harvard University Press, 1960), p. 120.
Lieutenant Colonel Herman L. Gilster (USMA; Ph.D., Harvard University) is assigned to the Directorate of Operations Analysis, Hq PACAF. Except for a year as Chief, Tactical Analysis Division, Hq Seventh Air Force, PACAF, he was on the faculty of USAF Academy, 196371, as Associate Professor of Economics and Management. His articles have been published in Papers in Quantitative Economics (University of Kansas Press), Air University Review, Operations Research, etc.
Disclaimer
The conclusions and opinions expressed in this
document are those of the author cultivated in the freedom of expression,
academic environment of Air University. They do not reflect the official
position of the U.S. Government, Department of Defense, the United States Air
Force or the Air University.
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