To prove to yourself that the periphery of the retina is more sensitive to light, go outdoors on a starry night. Wait for about 20 minutes to let your eyes get used to the dark, then look at a fairly bright star. While continuing to focus on this star, find another, dimmer one that you can just detect with your peripheral vision. Then turn your eyes until you are looking at this dimmer star directly, and you'll see it disappear!

The explanation is that when the image of the dimmer star was projected onto the periphery of your retinas, where rods predominate, their greater sensitivity to light just enabled them to detect it. But when you turned to look at the star directly, its image was projected onto the centres of your retinas, the foveas, where cones predominate. Your cones are less sensitive to light and so could not see the star at all.