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Click on the phrases to see them in context. The original texts by Immanuel Kant and David Hume are available from the Gutenberg Projet.

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The science of mathematics presents the most brilliant example of the extension of the sphere of pure reason without the aid of experience.

 SS 22. Transcendental Deduction of the universally possible employment in experience of the Pure Conceptions of the Understanding. 
When I received the relations of resemblance, contiguity and causation, as principles of union among ideas, without examining into their causes, it was more in prosecution of my first maxim, that we must in the end rest contented with experience, than for want of something specious and plausible, which I might have displayed on that subject.
 
It is from the prospect of pain or pleasure that the aversion or propensity arises towards any object: And these emotions extend themselves to the causes and effects of that object, as they are pointed out to us by reason and experience.
 A transcendental proposition is, therefore, a synthetical cognition of reason by means of pure conceptions and the discursive method, and it renders possible all synthetical unity in empirical cognition, though it cannot present us with any intuition a priorI. [*Footnote; In the case of the conception of cause, I do really go beyond the empirical conception of an event--but not to the intuition which presents this conception in concreto, but only to the time-conditions, which may be found in experience to correspond to the conception. But the schema which is originated from an idea (in which case reason presents us with aims a priori, and does not look for them to experience), forms the basis of architectonical unity. These examples from history and common experience are rich and curious; but we may find parallel ones in the arts, which are no less remarkable. But if the rationalist is bold enough to construct, on the mere authority of the faculty of thought--without any intuition, whereby an object is given--a self-subsistent being, merely because the unity of apperception in thought cannot allow him to believe it a composite being, instead of declaring, as he ought to do, that he is unable to explain the possibility of a thinking nature; what ought to hinder the materialist, with as complete an independence of experience, to employ the principle of the rationalist in a directly opposite manner-- still preserving the formal unity required by his opponent?] It follows from this that you must accept the absolutely necessary as out of and beyond the world, inasmuch as it is useful only as a principle of the highest possible unity in experience, and you cannot discover any such necessary existence in the would, the second rule requiring you to regard all empirical causes of unity as themselves deduced. Provides helpful tips on how, when and where to apply. For they are tantamount to asking whether all things as phenomena do without exception belong to the complex and connected whole of a single experience, of which every given perception is a part which therefore cannot be conjoined with any other phenomena--or, whether my perceptions can belong to more than one possible experience?