Explain, with examples, the ways in which the Renaissance spirit of a rediscovery of the classics and the new Humanist ways of thinking were expressed in the design and content of the Italian Renaissance garden.

The Renaissance era, particularly in Italy, brought about major changes in thought that influenced garden design. The two major components of the Renaissance school of thought were a rediscovery of classical texts and philosophies, and the humanist reevaluation of man’s relationship with God and nature (accompanied by the thought that nature was essential, but could be improved upon through human ordering and cultivation). These new emphases led to an age of human focus, and an increase in studies of the arts and sciences. 

The focus on man rather than the divine led to an opening of the garden, moving away from closed gardens of the Medieval era. Gardens became more expansive and outwardly directed, enabling open enjoyment by people. Additionally, gardens lost their separation from the home, instead becoming a much more integral and noticeable feature.

http://www.stuff.co.nz/marlborough-express/your-marlborough/home-and-garden/christo-saggers/4066844/Romantic-renaissance

New developments in math and science were reflected in the sequential, geometric array of garden form, giving gardens of the Italian Renaissance a very structured appearance, with a return of axial alignment and mirrored sides. Similarly, advances in technology led to an increase in elaborate water features in gardens, demonstrating the power and innovation of mankind.

http://en.wikipedia.org/wiki/Italian_Renaissance_garden#mediaviewer/File:Villa_d_Este1.jpg

The villa became a key feature of Italian Renaissance gardens, with the purpose of leisure, taking inspiration from the classics, “otium,” or creative pursuits. The villa was meant to be a central feature; both something to be admired and a place from which to admire gardens. Villas and their corresponding gardens were elaborate, meant as a show of wealth and power, further emphasizing human rather than religious motives. Within these gardens, family symbols, through statuary, water features, or other design, were another showing of status.

http://www.honeysucklelife.com/2012/03/05/image-gallery-formal-gardens/olympus-digital-camera-2/

The following link contains information about several well-known examples of Italian Renaissance garden art: 

http://www.gardenvisit.com/history_theory/library_online_ebooks/ml_gothein_history_garden_art_design/early_baroque_gardens_italy