Every time he bites, scratches, pushes, screams on purpose at the top of his lungs, throws a toy angrily, grabs, etc..... pick his little body up (to remove him and the other child from dangerous contact), look him in the eye and say in a deep sturn voice, "No hitting" or whatever the offense , then turn his little body away from you and place him on all fours (face down) on the floor. Don't smash him or be brutal... but make it unpleasent and quick. Doing this will give him social rejection rather than reiforcement for the aggressive behavior. The only thing he'll see is your sturn face, your voice, and the floor, rather than the other child crying over getting hurt or toy taken away. Stay on him.... be consistant. Spend your time at his side ready to remove him immediately. You'll find that you'll spend your time on your knees next to him a lot. Just keep telling him "No" and turning him away -over and -over and -over again. He won't like it.
But, it's not abusive, it's not spanking, and it's not over doing the social rejection like time outs tend to do. And at 15 months old... they can't link the two together. So, when you turn him away toward the floor, he'll probably cry. He may even tantrum, but don't hold him hostage to the position. Just leave him there alone. When he's done being upset over it, he'll join back in. He might need a hug after a minute or two if he really feels bad but, definately don't scoop him up and cottle him just because he feels bad for what he did. Give him a minute to just be sad about it.
It works.
Oh, and ofcourse.... spend all the rest of your time verbally encouraging him and reinforcing the good things that he does. Give him plenty of feed back saying stuff like, "Oh, you wanted Sally's toy but you didn't take it. You didn't hit. You kept your hands down. Look at you....you walked past the baby and did not poke her eye. Yes, you sit down by her and pat her belly gentle. Blah, Blah, Blah."
But, it works.