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The earliest references to hatha yoga are in Buddhist works dating from the eighth century. The earliest definition of hatha yoga is found in the 11th century Buddhist text Vimalaprabha, which defines it in relation to the center channel, bindu etc. Hatha yoga synthesizes elements of Patanjali's Yoga Sutras with posture and breathing exercises. It marks the development of asanas (plural) into the full body 'postures' now in popular usage and, along with its many modern variations, is the style that many people associate with the word yoga today.
The first Hindu teacher to actively advocate and disseminate aspects of yoga, not including asanas, to a western audience, Swami Vivekananda, toured Europe and the United States in the 1890s. The reception which Swami Vivekananda received built on the active interest of intellectuals, in particular the New England Transcendentalists, among them Ralph Waldo Emerson (1803–1882), who drew on German Romanticism and philosophers and scholars like G. W. F. Hegel (1770–1831), the brothers August Wilhelm Schlegel (1767–1845) and Karl Wilhelm Friedrich Schlegel (1772–1829), Max Mueller (1823–1900), Arthur Schopenhauer (1788–1860), and others who had (to varying degrees) interests in things Indian.
An influential text which teaches yoga from an Advaita perspective of nondualistic idealism is the Yoga-Vāsiṣṭha. This work uses numerous short stories and anecdotes to illustrate its main ideas. It teaches seven stages or bhumis of yogic practice. It was a major reference for medieval Advaita Vedanta yoga scholars and before the 12th century, it was one of the most popular texts on Hindu yoga.
Core: Yes. There are yoga poses to target just about every core muscle. Want to tighten those love handles? Then prop yourself up on one arm and do a side plank. To really burn out the middle of your abs, you can do boat pose, in which you balance on your "sit bones" (the bony prominences at the base of your pelvic bones) and hold your legs up in the air.