On The External Control Of Our Environments:
I considered how one and the same man with the very same mind, were he brought up from infancy among the French or the Germans, would become different from what he would be had he always lived among the Chinese or the cannibals, and how, even down to the styles of our clothing, the same thing that pleased us ten years ago, and that perhaps will again please us ten years hence, now seems to us extravagant and ridiculous.  … Hence I [chose to be] constrained to try to guide myself on my own.
(Discourse On Method, Part II, page 9-10)

The Method:
[Just as law is best kept dramatically parsimonous and rigidly enforced, so these principles]:
1) The first [principle] was to never accept anything as true that I did not plainly know to be such; that is to say, carefully to avoid hasty judgment than what presented itself to my mind so clearly and so distinctly that I had no occasion to call it in doubt.
2) The second [principle], to divide each of the difficulties I would examine into as many parts as possible, and as was required, in order better to resolve them.
3) The third [principle], to conduct my thoughts in an orderly fashion, by commencing with those objects that are simplest and easiest to know, in order to ascend little by little, as by degrees, to the knowledge of the most composite things…
4) And the last [principle], everywhere to make enumerations so complete and reviews so general that I was assured of having omitted nothing.
(Discourse On Method, Part II, page 11)

On Deriving Context From Graphical Models:
I thought it would be more worthwhile for me to examine only these [mathematical concepts] in general, and to suppose them only in subject that would help me make the knowledge of them easier…. Then, having noted that, in order to know these [concepts], I would sometimes need to consider each of them individually, and sometimes only to keep them in mind, or to grasp many of them together, I thought that, in order better to consider them in particular, I ought to suppose them to be relations between lines, since I found nothing more simple, or nothing that I could represent more distinctly to my imagination and to my senses; but that, in order to keep them in mind or to grasp many of them together, I would have to explicate them by means of certain symbols, the briefest ones possible; and that by this means I would be borrowing all that is best in geometrical analysis and algebra, and correcting all the defects of the one by the other.
(Discourse On Method, Part III, page 12)

On The Distinction Between Reason And Empiricism:
Everything unimaginable [to the senses] seems to [many people] unintelligible.  Even philosophers take it as a maxim in the schools that there is nothing in the understanding that has not first been in the senses, where it is nevertheless certain that the ideas of God and the soul have never been.  And it seems to me that those who want to use their imagination to grasp these ideas are doing the very same thing as if, in order to hear sounds or to smell odors, they wanted to use their eyes.
(Discourse On Method, Part IV, page 21)

On The Certainty Of Metaphysics Over The Sciences:
Finally, if there still are men who have not been sufficiently persuaded of the existence of God and of their soul by means of the reasons I have brought forward, I very much want them to know that all the other things of which they think themselves perhaps more assured, such as having a body, that there are stars and an earth, and the like, are less certain.
(Discourse On Method, Part IV, page 21)

On Truth Originating From Divine Clarity:
It follows from [divine perfection] that our ideas or notions, being real things and coming from God, cannot, in all that is clear and distinct in them, be anything but true.  Thus, if we quite often have ideas that contain some falsity, this can only be the case with respect to things that have something confused or obscure about them, because in this respect they participate in nothing; that is, they are thus confused in us only because we are not perfect…. But if we did not know that all that is real and true in us comes from a perfect and infinite being – however clear and distinct our ideas were, we would have no reason that assured us that they had the perfection of being true.  
(Discourse On Method, Part IV, page 22)

On Philosophical Zombies:
I contented myself with supposing that God formed the body of a man exactly like one of ours, as much in the outward shape of its members as in the internal arrangement of its organs… without putting into it, at the start, any rational soul, or anything else to serve there as a vegetative or sensitive soul, but merely kindled in the man’s heart one of those fires without light….  For on examining the functions that could, as a consequence, be in this body, I found there precisely all those things that can be in us without our thinking about them, and hence, without our soul’s contributing to them….  And these are all the same features in which one can say that animals lacking reason resemble us.
(Discourse On Method, Part V, page 26)

[While this is not the case for animals], if there were any such machines that bore a resemblance to our bodies and imitated our actions as far as this is practically feasible, we would always have two very certain means of recognizing that they were not at all true men.  The first is that they could never use words or other signs, or put them together as we do in order to declare our thoughts to others.  For one can well conceive of a machine that utters words, and even that utters words appropriate to the bodily actions that will cause some change in its organs….  But it could not arrange its words differently so as to respond to the sense of all that will be said in its presence, as even the dullest man can do.  The second means is that, although they might perform many tasks very well or perhaps better than any of us, such machines would invitably fail in other tasks; by this means one would discover that they were acting not through knowledge but only through the disposition of their organs.  For while reason is universal… these organs require some particular disposition for each particular action; consequently, it is for all practical purposes impossible for there to be enough different organs in a machine to make it act in all the contingencies of life in the same way as our reason makes us act.  
(Discourse On Method, Part VI, page 32)

On The Purpose Of The Lungs:
The true function of respiration is to bring enough fresh air into the lungs to cause the blood which comes there from the right cavity of the heart, where it has been rarified and, as it were, changed into vapors, immediately to be condensed and to be converted once again into blood before returning to the left cavity; without this process the blood could not properly aid in feeding the fire that is in the heart.
(Discourse On Method, Part V, page 30)
[Note: this was, of course, ultimately proved incorrect]

On Immortality Of The Soul:
When one knows how different [soul and body] are, one understands much better the arguments which prove that our soul is of a nature entirely independent of the body, and consequently that it is not subject to die with it.  Then, since we do not see any other causes at all for its destruction, we are naturally led to judge from this that it is immortal.  
(Discourse On Method, Part VI, page 33)

On The Inferiority Of A-T Philosophy Against Science:
It is possible to arrive at knowledge that would be very useful in life and that, in place of that speculative philosophy taught in the schools , it is possible to find a practical philosophy, by means of which, knowing the force and the actions of fire, water, air, the stars, the heavens, and all the other bodies that surround us, just as distinctly as we know the various skills of our craftsmen, we might be able, in the same way, to use them for all the purposes for which they are appropriate, and thus render ourselves, as it were, masters and possessors of nature… perhaps even [rid oneself] of the frailty of old age…
(Discourse On Method, Part VI, page 35)
[Note: “philosophy taught in the schools” is likely directed at A-T scholasticism]

I am sure that that the most impassioned of those who now follow Aristotle would believe themselves fortunate, if they had as much knowledge of nature as he had, even if it were on the condition that they would never have any more.  They are like ivy, which never stretches any higher than the trees supporting it, and which often even descends again after if has reached their tops, for it seems to me that they are re-descending, that is, they are making themselves somehow less knowledgeable than if they abstained from studying; not content with knowing all that is intelligibly explained in their author, they want in addition to find the solutions there to many difficulties about which he says nothing and about which he has perhaps never thought.  Still, their manner of philosophizing is very convenient for those who have only very mediocre minds, for the obscurity of the distinctions and the principles they make use of is the reason why they can speak about all things as boldly as if they knew them, and why they can uphold everything they say against the most subtle and the most adroit, without anyone’s having the means of convincing them they are mistaken.  In this they seem to me like a blind man who, in order to fight without a disadvantage against someone who is sighted, had made his opponent go into the depths of some very dark cellar.  
(Discourse On Method, Part VI, page 40)

On His Aspirations For Inspiring Post-Mortem Audiences:
I [published] as much to have all the more occasion to examine [my judgments] well (since without doubt one always looks more carefully at what one believes must be seen by many, than at what one does only for oneself; and often the things that have seemed to me to be true when I began to conceive them have appeared false to me when I wanted to put them on paper), as in order not to lose any occasion to benefit the public, if I am able, and in order that, if my writings are worth anything, those who will have them after my death can thus use them as will be most fitting.
(Discourse On Method, Part VI, page 37)

On Reasons For Self-Censorship After Galileo:
I would have many opportunities to lose time, had I published the foundations of my physics.  For although they are nearly all so evident that it is necessary only to understand them in order to believe them, and although there has not been a single one for which I did not believe I could give demonstrations, nevertheless, because it is impossible for them to be in agreement with all the diverse opinions of other men, I foresee that I would often be distracted by the disputes they would engender.
(Discourse On Method, Part VI, page 38)

On The Disadvantages Of Social Learning:
In this way, if there were someone in the world whom one assuredly knows to be capable of finding the greatest things and the things as beneficial to the public as possible and whom, for this cause, other men were to exert themselves to help in every way to succeed in his plans, I do not see that they could do a thing for him except to make a donation toward the expenses of the experiments he would need and, for the rest, to prevent his leisure from being wasted by the importunity of anyone.
(Discourse On Method, Part VI, page 41)

On Experience Being The Groundwork Of Semantic Webs:
The reasonings [in my essay presenting a considerable number of scientific results] follow each other in such a way, just as the last are demonstrated by means of the first, which are their causes, so these first are reciprocally demonstrated by means of the last, which are their effects.  And one must not imagine that I am here committing the fallacy that logicians call a circle, for, experience rendering the majority of these effects very certain, the causes from which I deduce these effects serve not so much to prove them as to explain them; on the contrary, it is rather the case that the causes are what are proved by the effects.
(Discourse On Method, Part VI, page 43)