Chronological Order vs. Order of Exposition

Strauss's circular structure (going clockwise), in which the first (Heidegger) and last (Burke) chapter are thematically grouped, is connected by the hinge of the introduction, creating a Burke-Intro-Heidegger dyad on top mirrored by an Aristotle-Intro-Hobbes dyad on bottom.

Pairs in Natural Right and History

This is a schema of Natural Right and History. Heidegger-Burke-Rousseau-Plato are lined up under philosophy and Weber-Locke-Hobbes-Aristotle are under science. Each quadrant consists of a pair with an identical number of paragraphs - Heidegger-Burke (34 paragraphs), Weber-Locke (42 paragraphs), Rousseau-Plato (44 paragraphs), and Hobbes-Aristotle (40 paragraphs). Rousseau-Plato actually don't have the same number of paragraphs (44 and 48, respectively), but the last 4 paragraphs of Plato, which treat egalitarianism (in its ancient and modern types), belongs equally with either the Plato or Rousseau chapters (see 3.1, at bottom).

The way to think about the distinction between the "idea of philosophy" and the "idea of science" is that Hobbes-Aristotle are natural philosophers with positive doctrines, Weber-Locke reject natural philosophy but are methodological naturalists, Heidegger-Burke reject the possibility of philosophy (and science), and Rousseau-Plato are philosophers. Another way to look at it is that the top two quadrants reject intelligibility and the figures on the bottom assume some level of intelligibility. In that sense Rousseau and Hobbes are not typical moderns, hence the pairing with Plato and Aristotle, respectively.

For more discussion of pairs, see here. And for a discussion of intelligibility as the dividing line between ancient and modern, see here.

Cato appears three times in NRH. Cato may be the embodiment of non-philosophic virtue, as conceived by Strauss. Here are the passages:

p. 168: By identifying traditional political philosophy with the idealistic tradition, Hobbes expresses, then, his tacit agreement with the idealistic view of the function or scope of political philosophy. Like Cicero before him, he sides with Cato against Carneades. 

That passage is explained by this later passage:

p. 196: The principle which was supposed to make possible a political doctrine of universal applicability, then, is not universally valid and therefore is useless in what, from Hobbes point of view, is the most important case-the extreme case. For how can one exclude the possibility that precisely in the extreme situation the exception will prevail.*

* Leviathan XIII and XV. One may state this difficulty also as follows: In the spirit of the dogmatism based on skepticism, Hobbes identified what the skeptic Carneades apparently regarded as the conclusive refutation of the claims raised on behalf of justice, with the only possible justification of these claims: the extreme situation-the situation of two shipwrecked men on a plank on which only one man can save himself-reveals not the impossibility of justice, but the basis of justice. Yet Carneades did not contend that in such a situation one is compelled to kill one's competitor (cf. Republic iii. 29-30): the extreme situation does not reveal a real necessity. 

Cato appears again in the Rousseau chapter:
p. 255: "Rousseau indicates the meaning of virtue clearly enough for his purpose by referring to the examples of the citizen-philosopher Socrates, Fabricius, and, above all, of Cato: Cato was the "greatest of men."*
* First Discourse, p. 59: Cato has given the human race the spectacle and model of the purest virtue which has ever existed.

At the end of the Rousseau chapter, here is what Strauss says:
p. 294: "As he confessed at the end of his career, no book attracted and profited him as much as the writings of Plutarch. The solitary dreamer still bowed to Plutarch's heroes," i.e. Cato.

And then again in Burke:
p. 318: "It is only a short step from this thought of Burke to the supersession of the distinction between good and bad by the distinction between the progressive and retrograde, or between what is and what is not in harmony with the historical process. We are here certainly at the pole opposite to Cato, who dared to espouse a lost cause."

Strauss himself, who espouses a lost cause, identifies to some extent with Cato (and Cicero), and with Cato (Cicero) against Carneades. Strauss agrees with the idea that "the extreme situation does not reveal a real necessity."

The argument for capitalism (and ultimately historicism) is a syllogism that looks like this:

Major Premise) We have a moral duty to contribute to the common good

Minor Premise) Limitless accumulation contributes to the common good

ERGO) Limitless accumulation is a moral duty

This argument leads to historicism because the fundamental premise is that processes which are unguided (limitless accumulation) lead to ends (the common good) which are unplanned.

NRH, p. 315: "The good order or the rational or the rational is the result of forces which do not tend to the good order or the rational. The principal was first applied to the planetary system and then to the system of wants, i.e. to economics."*

* Cf. Hegel, Rechtsphilosophie, sec. 189 Zusatz.

"Political economy is the science which starts from this view of needs and labour but then has the task of explaining mass-relationships and mass-movements in their complexity and their qualitative and quantitative character. This is one of the sciences which have arisen out of the conditions of the modern world. Its development affords the interesting spectacle (as in Smith, Say, and Ricardo) of thought working upon the endless mass of details which confront it at the outset and extracting therefrom the simple principles of the thing, the Understanding effective in the thing and directing it. . . . The most remarkable thing here is this mutual interlocking of particulars, which is what one would least expect because at first sight everything seems to be given over to the arbitrariness of the individual, and it has a parallel in the solar system which displays to the eye only irregular movements, though its laws may none the less be ascertained."

To find this same sort of argument in Newton, see the General Scholium to Book III, The System of the World, in the Principia.

"Wild mentions the following characteristic traits of "idealism": the subordination of ontology to logic (p.2); the denial of the intentionality of thought (280 n. 301); the view "that all things are constantly thinking, or that there are unconscious or non-thinking thoughts" (214); the confusion of material things with the forms, the objects of thought, and hence the denial of matter, motion, and change (5,234, 238, 290). His last word on the subject is the identification of idealism with "the confusion of man with the creator" (301), that is, with the view that all meaning, order and truth are originated by, or relative to, "consciousness," "reason," "the subject," "man," or Existenz. (Compare Edmund Husserl, Ideen, $$ 47, 49, 55, and M. Heidegger, Sein und Zeit, $ 44, as well as "Vom Wesen des Grundes" in Festschrift fur Edmund Husserl, Halle 1929, pp. 98 ff.)

I refer to Husserl and Heidegger because they most clearly reveal that Wild's identification of idealism with the denial of intentionality or with the subordination of ontology to logic does not go to the root of the matter. Wild's position is at least as much opposed to English empiricism, for example, as it is to German idealism. Yet he has chosen to present German idealism as the villian. A man who claims to be a Platonist is under an obligation to stress the fact that German idealism attempted to restore important elements of Plato's and Aristotle's teaching in opposition to western (English and French) philosophy, if on the basis of of a foundation laid by Western philosophy."

"On a New Interpretation of Plato's Political Philosophy," Leo Strauss, Social Research, September 1946, 335-6

Batnitzky makes an interesting argument that the law is for Strauss the limit that keeps philosophy from trying to encompass everything, including politics. Philosophy was preserved in Muslim countries precisely because philosophy was limited by the law. This part of the review is interesting:

"The Scholastics, culminating in Thomas Aquinas, claimed that by argument they could show the rationality of religion. True enough, some doctrines, such as the Trinity, are knowable only through revelation; but at least the existence and principal attributes of God can be established by argument from premises not in doubt. For all this, Strauss had no use: he viewed Scholasticism as not only false but dangerous. Much more to his liking was Averroism, which taught that philosophy and religion were competing truths.2"In contrast to the Islamic-Jewish world, Strauss claims, the melding of revelation and philosophy in medieval Christendom destroyed the meanings of both revelation and philosophy.3In a very important sense, Strauss seems to locate the invention of the possibility of an atheistic, secular society with Thomas Aquinas. . . " (B, p.122)

Strauss of course didn't write much on Aquinas, but if you want to know where you can find this argument, it can be found in Jaffa's Thomism and Aristotelianism. The problem with Aquinas, according to Jaffa, is that he is a) too precise regarding matters that do not allow of precision and b) he identifies conscience with intuition and c) is also too pessimistic about the possibility of philosophy. I'm assuming Jaffa is transcribing a course on Aquinas that he took at the New School from Strauss, or developing an argument of Strauss's. Jaffa also touches on Aquinas' critique of "Averroism," and seems to argue on behalf of the Averroists against Aquinas.

Strauss on Husserl

"The decisive point in Husserl is the critique of modern science in the light of genuine science, that is to say, Platonic-Aristotelian. His work can only be understood in the light of the enormous difficulties in which Platonic-Aristotelian culminated, namely, the problem of nous. Considering the enormous difficulties of understanding de anima iii.5, Husserl's egological foundation of the ontologies is at least excusable."

Letter to Voegelin, October 11, 1943

"On the question of modern philosophy and progress: modern philosophy (or science) is originally the attempt to replace the allegedly or really inadequate classical (and that means, at the same time, medieval) philosophy (or science) by the correct philosophy. The inadequacy was this: the achieved science of antiquity (Plato and Aristotle) was not capable of giving an account of certain natural phenomena (of the external world) which on its own terms it had to give an account of. The idea arose that the materialistic physics, displaced by classical philosophy, that is, above all by the Aristotelian physics, offered an unheard of expansion of the possibilities of knowledge. But: one had learned from Plato-Aristotle that a materialistic physics cannot understand itself, the possibility of knowledge (noein). Thus the task: first to secure the possibility of knowledge, in order then to be able to proceed with mechanistic physics, and so to be able to understand the universe. That is the meaning of Descartes Meditations, of the fundamental book of modern philosophy."

Letter to Lowith, August 20, 1946